#6539807

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	#39356	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 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)(13,14,19,23)(15,20,26,35)(16,27,29,37)(17,22,18,31)(21,32,38,28)(24,39,41,44)(25,34,30,36)(33,40,43,42);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,30)(18,25)(19,26)(21,33)(22,36)(23,35)(27,39)(28,42)(29,41)(31,34)(32,40)(37,44)(38,43);(1,3,4,2)(6,9)(10,12)(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,37)(23,38)(25,40)(26,41)(27,31)(30,42)(34,39)(35,43)(36,44);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,31)(15,25,26,30)(16,28,29,32)(20,34,35,36)(21,37,38,27)(24,40,41,42)(33,39,43,44);(7,11)(10,12)(15,26)(20,35)(24,41)(25,30)(33,43)(34,36)(39,44)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6447003);