#503426

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	C₂×#3285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1000 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)(7,11)(8,10)(9,12)(13,14,20,25)(15,21,28,41)(16,29,27,42)(17,23,19,36)(18,34,35,44)(22,37,40,26)(24,30,38,31)(32,43,33,39);(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,30)(17,26)(18,33)(19,37)(20,28)(22,23)(24,42)(25,41)(27,31)(29,38)(32,35)(34,39)(36,40)(43,44);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,16,31,28,19,32,35,37,20,27,30,15,17,33,18,26)(14,22,34,39,23,21,38,42,25,40,44,43,36,41,24,29);(5,8,9,7)(6,11,12,10)(13,17,20,19)(14,23,25,36)(15,26,28,37)(16,32,27,33)(18,30,35,31)(21,22,41,40)(24,34,38,44)(29,43,42,39)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=efh,d^2=ei,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, 410622);