#6552538

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	#38095	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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)(13,14,17,21,18,22,31,37)(15,23,26,39,25,19,32,34)(16,27,30,36,29,35,28,20)(24,40,43,33,41,44,42,38);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,33)(21,34)(22,23)(27,44)(28,42)(29,41)(30,43)(31,32)(35,40)(36,38)(37,39);(1,3,4,2)(6,9)(10,12)(13,16,18,29)(14,20,22,36)(15,24,25,41)(17,28,31,30)(19,33,23,38)(21,35,37,27)(26,42,32,43)(34,40,39,44);(5,7)(6,10)(8,11)(9,12)(15,25)(19,23)(24,41)(26,32)(33,38)(34,39)(40,44)(42,43);(7,11)(10,12)(13,17,18,31)(14,21,22,37)(15,26,25,32)(16,28,29,30)(19,34,23,39)(20,35,36,27)(24,42,41,43)(33,40,38,44)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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, 6459734);