#6552326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#38948	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1839 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)(13,14,17,21,18,22,32,38)(15,23,27,34,26,19,25,35)(16,28,31,37,30,36,29,20)(24,40,41,33,42,44,43,39);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,22,23)(16,24,30,42)(17,27,32,25)(20,33,37,39)(21,35,38,34)(28,44,36,40)(29,43,31,41);(1,3,4,2)(6,9)(10,12)(13,16,18,30)(14,20,22,37)(15,24,26,42)(17,29,32,31)(19,33,23,39)(21,36,38,28)(25,41,27,43)(34,44,35,40);(5,7)(6,10)(8,11)(9,12)(13,17,18,32)(14,21,22,38)(15,25,26,27)(16,29,30,31)(19,34,23,35)(20,36,37,28)(24,41,42,43)(33,44,39,40);(7,11)(10,12)(15,26)(19,23)(24,42)(25,27)(33,39)(34,35)(40,44)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459522);