#6467586

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2293 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)(8,9)(11,12)(13,14,17,21,18,22,31,38)(15,23,26,32,24,37,30,19)(16,20,29,36,28,35,43,44)(25,39,40,34,27,41,42,33);(2,3)(13,15)(14,19)(16,27)(17,30)(18,24)(20,34)(21,37)(22,32)(23,38)(25,28)(26,31)(29,40)(33,35)(36,39)(41,44)(42,43);(1,2,4,3)(6,9)(10,12)(15,24)(19,32)(23,37)(25,27)(26,30)(33,34)(39,41)(40,42);(1,3,4,2)(13,16)(14,20)(15,25)(17,29)(18,28)(19,33)(21,36)(22,35)(23,39)(24,27)(26,40)(30,42)(31,43)(32,34)(37,41)(38,44);(5,7,8,11)(6,10,9,12)(16,28)(20,35)(25,27)(29,43)(33,34)(36,44)(39,41)(40,42)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374782);