#493842

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1116 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401038);