#510012

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

564 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,9,12)(7,8,11,10)(13,14,20,35,15,26,33,22,17,24,18,25,19,21,28,39)(16,29,44,37,30,42,34,41,31,38,32,40,27,43,36,23);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15,17,19)(14,21,24,26)(16,30,31,27)(18,28,20,33)(22,35,39,25)(23,40,41,37)(29,43,38,42)(32,36,44,34);(2,4)(5,8,9,10)(6,11,12,7)(14,22)(15,19)(18,33)(20,28)(21,25)(23,38)(24,39)(26,35)(27,30)(29,41)(32,34)(36,44)(37,42)(40,43);(6,12)(7,11)(13,16,17,31)(14,23,24,41)(15,27,19,30)(18,34,20,36)(21,37,26,40)(22,38,39,29)(25,42,35,43)(28,44,33,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 417208);