#506524

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

786 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)(13,14,20,37,35,27,21,28,18,23,19,24,17,15,25,38)(16,29,43,41,31,30,36,22,32,44,33,26,34,42,39,40);(5,7,8,12)(6,9,10,11)(13,15,20,38,35,14,21,37,18,27,19,28,17,23,25,24)(16,30,43,22,31,44,36,26,32,42,33,40,34,29,39,41);(2,4)(13,16)(14,22)(15,26)(17,31)(18,32)(19,36)(20,39)(21,33)(23,40)(24,42)(25,43)(27,41)(28,29)(30,37)(34,35)(38,44);(6,10)(9,11)(13,17,18,35)(14,15,23,27)(16,31,32,34)(19,21,20,25)(22,26,40,41)(24,28,37,38)(29,30,44,42)(33,39,43,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 413720);