#6576474

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#36078	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2736 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:

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

Polycyclic presentation:

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

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, 6483670);