#346496

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62887	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2295 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2295 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(5,8)(9,10,16,22)(11,23,27,44)(12,28,31,46)(13,19,33,43)(14,21,15,20)(17,34,39,26)(18,36,41,30)(24,37,52,63)(25,53,56,48)(29,58,61,65)(32,45,35,42)(38,55,64,54)(40,59,66,60)(47,50,57,51)(49,68,70,62)(67,69,72,71);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,24)(13,25)(14,34)(15,26)(16,27)(18,37)(19,38)(20,44)(21,23)(22,39)(28,57)(29,49)(30,51)(31,52)(32,55)(33,56)(35,54)(36,50)(40,62)(41,63)(42,48)(43,64)(45,53)(46,47)(58,72)(59,71)(60,69)(61,70)(65,67)(66,68);(1,3,4,7)(2,5,6,8)(9,12)(10,18)(11,24)(13,29)(14,30)(15,36)(16,31)(17,37)(19,40)(20,28)(21,46)(22,41)(23,47)(25,49)(26,50)(27,52)(32,59)(33,61)(34,51)(35,60)(38,62)(39,63)(42,65)(43,66)(44,57)(45,58)(48,67)(53,72)(54,69)(55,71)(56,70)(64,68);(9,13,16,33)(10,19,22,43)(11,25,27,56)(12,29,31,61)(14,35,15,32)(17,38,39,64)(18,40,41,66)(20,45,21,42)(23,48,44,53)(24,49,52,70)(26,55,34,54)(28,58,46,65)(30,60,36,59)(37,62,63,68)(47,67,57,72)(50,71,51,69)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253692);