#502897

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×#2250	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

888 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)(8,9)(10,12)(13,14)(15,29)(16,33)(17,28)(18,25)(19,39)(20,40)(21,24)(22,31)(23,37)(26,35)(27,34)(30,41)(32,44)(36,43)(38,42);(6,9)(10,11)(13,15,18,31)(14,22,25,29)(16,34,32,35)(17,30,21,37)(19,36,20,38)(23,24,41,28)(26,44,27,33)(39,42,40,43);(1,2,3,4)(13,16,35,15,21,36,20,30,18,32,34,31,17,38,19,37)(14,23,39,42,28,22,27,44,25,41,40,43,24,29,26,33);(5,7,8,12)(6,10,9,11)(13,17,18,21)(14,24,25,28)(15,30,31,37)(16,36,32,38)(19,34,20,35)(22,41,29,23)(26,40,27,39)(33,42,44,43)

Polycyclic presentation:

2^9|b^2=i,c^2=fh,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, 410093);