#6456577

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1865 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)(15,19,26,35)(16,20,28,37)(17,21,31,40)(23,32,41,43)(24,33,25,34)(27,36,42,44)(29,38,30,39);(2,3)(13,15)(14,19)(16,27)(17,24)(18,26)(20,36)(21,33)(22,35)(23,29)(25,31)(28,42)(30,41)(32,38)(34,40)(37,44)(39,43);(1,2,4,3)(6,9)(10,12)(13,16,18,28)(14,20,22,37)(15,23,26,41)(17,30,31,29)(19,32,35,43)(21,39,40,38)(24,27,25,42)(33,36,34,44);(1,3,4,2)(15,24)(17,31)(19,33)(21,40)(23,27)(25,26)(29,30)(32,36)(34,35)(38,39)(41,42)(43,44);(5,7,8,11)(6,10,9,12)(16,28)(20,37)(23,41)(27,42)(29,30)(32,43)(36,44)(38,39)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,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, 6363773);