#346995

order	512
nilpotency class	3
isoclinism	isoclinic class of #346950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2037 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 2037 conjugacy classes of subgroups — too many to draw. Only its 222 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254191);