#6811640

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,12)(9,11)(13,14)(15,24)(16,28)(17,22)(18,20)(19,26)(21,29)(23,31)(25,33)(27,34)(30,36)(32,40)(35,41)(37,38)(39,42)(43,44);(2,3)(5,7,10,6)(8,12,9,11)(13,15)(14,19)(16,25)(17,26)(18,27)(20,32)(21,33)(22,24)(23,34)(28,37)(29,38)(30,39)(31,40)(35,42)(36,43)(41,44);(1,2,4,3)(14,20)(18,31)(19,32)(21,35)(22,23)(24,34)(27,40)(28,36)(30,41)(33,42)(37,43)(39,44);(5,8,10,9)(6,11,7,12)(13,16,17,29)(14,21,22,28)(15,25,26,38)(18,30,31,41)(19,33,24,37)(20,35,23,36)(27,39,40,44)(32,42,34,43);(1,3,4,2)(5,9,10,8)(6,11,7,12)(15,26)(16,29)(19,24)(21,28)(27,40)(30,41)(32,34)(35,36)

Polycyclic presentation:

2^9|b^2=fi,c^2=h,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718836);