#512327

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

578 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 162 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)(15,26,28,32)(16,31,23,19)(17,33,35,24)(20,36,34,25)(21,37,38,27)(29,40,44,41)(30,39,43,42);(5,7,8,11)(6,9,10,12)(13,15,19,29,20,30,17,27,18,28,31,44,34,43,35,37)(14,21,24,39,25,40,23,26,22,38,33,42,36,41,16,32);(6,10)(9,12)(14,22)(16,23)(21,38)(24,33)(25,36)(26,32)(39,42)(40,41);(2,4)(13,16,17,25,34,33,19,14,18,23,35,36,20,24,31,22)(15,26,27,41,43,39,29,38,28,32,37,40,30,42,44,21)

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,d^c=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, 419523);