#347168

order	512
nilpotency class	3
isoclinism	isoclinic class of #347154
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	#62886	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2145 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2145 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,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,e^c=ei,f^c=fi,g^c=gh,f^d=fh

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, 254364);