#512054

order	512
nilpotency class	4
isoclinism	isoclinic class of #511741
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1070 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,f^c=fh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419250);