#506528

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

840 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)(13,14,18,25,17,24,37,42,21,28,19,26,20,27,34,43)(15,22,30,23,16,33,35,44,32,40,31,39,29,38,36,41);(5,7,8,12)(6,9,10,11)(13,15,19,31,17,16,34,36,21,32,18,30,20,29,37,35)(14,22,26,39,24,33,43,41,28,40,25,23,27,38,42,44);(2,4)(13,16)(14,23)(15,17)(18,30)(19,31)(20,32)(21,29)(22,25)(24,41)(26,40)(27,44)(28,39)(33,43)(34,35)(36,37)(38,42);(6,10)(9,11)(13,17,21,20)(14,24,28,27)(15,29,32,16)(18,37,19,34)(22,38,40,33)(23,41,39,44)(25,42,26,43)(30,36,31,35)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,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, 413724);