#503419

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

995 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)(15,25)(16,29)(17,22)(18,35)(19,37)(20,26)(21,28)(23,30)(24,34)(27,44)(31,39)(32,40)(33,41)(36,43)(38,42);(5,7,9,8)(6,10,12,11)(13,15,17,26)(14,20,22,25)(16,30,27,31)(18,36,33,32)(19,38,34,28)(21,24,42,37)(23,29,39,44)(35,40,41,43);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,16,31,15,19,32,18,28,17,27,30,26,34,36,33,38)(14,21,41,40,24,25,23,29,22,42,35,43,37,20,39,44);(5,8,9,7)(6,11,12,10)(14,22)(20,25)(21,42)(23,39)(24,37)(29,44)(35,41)(40,43)

Polycyclic presentation:

2^9|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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410615);