#509985

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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 ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

628 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,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, 417181);