#6506687

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3090 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)(13,14,19,23)(15,24,28,39)(16,21,31,37)(17,22,18,32)(20,26,34,27)(25,38,42,43)(29,35,30,36)(33,40,44,41);(5,6)(7,11)(8,10)(9,12)(13,15,18,27,19,28,17,26)(14,20,22,24,23,34,32,39)(16,25,30,41,31,42,29,40)(21,33,35,38,37,44,36,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,25)(17,29)(18,30)(19,31)(20,33)(22,35)(23,37)(24,38)(26,40)(27,41)(28,42)(32,36)(34,44)(39,43);(7,12)(9,11)(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,29,31,30)(20,24,34,39)(21,35,37,36)(25,40,42,41)(33,38,44,43);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,30,31,29)(20,24,34,39)(21,36,37,35)(25,41,42,40)(33,43,44,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,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, 6413883);