#6465758

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=ghi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372954);