#93384

order	512
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#333
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂²
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

390 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 2 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,e^2=h,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=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, 580);