#503539

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

758 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=efhi,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410735);