#6467909

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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:

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

Polycyclic presentation:

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

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, 6375105);