#501491

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

645 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)(7,11)(8,9)(10,12)(15,24)(20,35)(25,41)(26,28)(27,36)(32,43)(37,38)(40,44);(6,9)(10,11)(13,14,17,22)(15,25,24,41)(16,21,19,34)(18,31,30,42)(20,36,35,27)(23,29,39,33)(26,40,28,44)(32,38,43,37);(2,4)(13,15,17,24)(14,20,22,35)(16,28,19,26)(18,32,30,43)(21,38,34,37)(23,40,39,44)(25,42,41,31)(27,29,36,33);(5,7,8,12)(6,10,9,11)(13,16,17,19)(14,21,22,34)(15,26,24,28)(18,29,30,33)(20,37,35,38)(23,31,39,42)(25,40,41,44)(27,43,36,32)

Polycyclic presentation:

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

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, 408687);