#471583

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1135 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=egh,c^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378779);