#6506750

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3049 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=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, 6413946);