#503545

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

768 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=efh,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410741);