#6456617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363813);