#6500080

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3181 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)(13,14,18,24,19,23,17,22)(15,25,27,34,28,38,32,20)(16,21,31,36,30,37,29,35)(26,39,40,43,41,44,42,33);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,26)(17,27)(18,32)(19,28)(21,33)(22,25)(23,34)(24,38)(29,40)(30,41)(31,42)(35,39)(36,44)(37,43);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,29)(18,31)(19,30)(20,33)(22,35)(23,37)(24,36)(25,39)(27,40)(28,41)(32,42)(34,43)(38,44);(7,12)(9,11)(13,17,19,18)(14,22,23,24)(15,27,28,32)(16,29,30,31)(20,25,34,38)(21,35,37,36)(26,40,41,42)(33,39,43,44);(6,10)(9,11)(14,23)(16,30)(20,34)(22,24)(25,38)(26,41)(29,31)(40,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407276);