#6807140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2000 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,24,34)(16,26,28,38)(17,30,32,21)(20,31,37,29)(23,39,27,44)(25,42,41,35)(33,43,36,40);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15,18,24)(14,19,22,34)(16,27,28,23)(17,25,32,41)(20,36,37,33)(21,35,30,42)(26,44,38,39)(29,43,31,40);(1,2,4,3)(13,16)(14,20)(15,23)(17,31)(18,28)(19,33)(21,38)(22,37)(24,27)(25,43)(26,30)(29,32)(34,36)(35,44)(39,42)(40,41);(6,10)(9,11)(15,24)(19,34)(23,27)(25,41)(33,36)(35,42)(39,44)(40,43);(1,3,4,2)(7,12)(9,11)(15,24)(16,28)(19,34)(20,37)(25,41)(26,38)(29,31)(35,42)

Polycyclic presentation:

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

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, 6714336);