#512324

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 608 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=efh,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=ghi,h^a=hi,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419520);