#6810859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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:

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

Polycyclic presentation:

2^9|a^2=i,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,d^b=di,e^b=efh,g^c=gi

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, 6718055);