#6432333

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2278 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)(8,9)(11,12)(13,14,17,21,18,22,30,36)(15,19,25,33,24,32,40,43)(16,26,27,34,28,35,29,20)(23,37,39,42,38,44,41,31);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,31)(21,33)(22,32)(26,37)(27,39)(28,38)(29,41)(30,40)(34,42)(35,44)(36,43);(2,4)(13,16,18,28)(14,20,22,34)(15,23,24,38)(17,29,30,27)(19,31,32,42)(21,35,36,26)(25,41,40,39)(33,44,43,37);(5,7,8,11)(6,10,9,12);(15,24)(19,32)(23,38)(25,40)(31,42)(33,43)(37,44)(39,41)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339529);