#6552628

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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:

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

Polycyclic presentation:

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

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, 6459824);