#512312

order	512
nilpotency class	4
isoclinism	isoclinic class of #511960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

551 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,5)(4,6)(7,8)(9,10,15,20)(11,21,26,48)(12,27,31,40)(13,32,34,64)(14,35,33,19)(16,36,39,24)(17,28,42,29)(18,43,44,22)(23,51,55,68)(25,57,56,47)(30,63,61,60)(37,66,67,54)(38,49,46,65)(41,62,59,50)(45,52,58,53)(69,72,71,70);(1,3,4,7)(2,5,6,8)(9,11,22,49,14,25,34,36,15,26,43,65,33,56,13,24)(10,16,32,47,19,38,44,48,20,39,64,57,35,46,18,21)(12,23,50,70,30,54,29,53,31,55,62,72,61,66,28,52)(17,37,63,69,41,51,27,58,42,67,60,71,59,68,40,45);(2,6)(5,8);(9,12,13,28,33,61,43,62,15,31,34,29,14,30,22,50)(10,17,18,40,35,59,64,60,20,42,44,27,19,41,32,63)(11,23,24,52,56,66,65,72,26,55,36,53,25,54,49,70)(16,37,21,45,46,68,57,71,39,67,48,58,38,51,47,69)

Polycyclic presentation:

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

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, 419508);