#506542

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

811 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)(8,10)(9,12)(13,14,19,26,17,24,33,43,21,28,18,25,20,27,37,42)(15,22,30,23,36,41,34,44,31,39,29,38,16,32,35,40);(5,7,8,12)(6,9,10,11)(13,15,18,29,17,36,37,35,21,31,19,30,20,16,33,34)(14,22,25,38,24,41,42,40,28,39,26,23,27,32,43,44);(2,4)(13,16,21,36)(14,23,28,38)(15,17,31,20)(18,30,19,29)(22,25,39,26)(24,40,27,44)(32,42,41,43)(33,34,37,35);(6,10)(9,11)(13,17,21,20)(14,24,28,27)(15,16,31,36)(18,37,19,33)(22,32,39,41)(23,40,38,44)(25,42,26,43)(29,34,30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413738);