#6811687

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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
prime radical	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718883);