#6597053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#37659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2208 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)(7,10)(8,11)(9,12)(13,14,19,23)(15,24,28,38)(16,21,29,35)(17,22,18,32)(20,26,34,27)(25,37,39,44)(30,36,31,42)(33,40,43,41);(5,7)(6,9)(8,12)(10,11)(13,15,18,27,19,28,17,26)(14,20,22,24,23,34,32,38)(16,25,31,41,29,39,30,40)(21,33,36,37,35,43,42,44);(2,4)(7,12)(9,10)(13,16)(14,21)(15,25)(17,30)(18,31)(19,29)(20,33)(22,36)(23,35)(24,37)(26,40)(27,41)(28,39)(32,42)(34,43)(38,44);(5,7)(6,10)(8,12)(9,11)(16,29)(21,35)(25,39)(30,31)(33,43)(36,42)(37,44)(40,41);(7,12)(9,10)(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,30,29,31)(20,24,34,38)(21,36,35,42)(25,40,39,41)(33,37,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504249);