#509277

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

583 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ei,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416473);