#93022

order	512
nilpotency class	4
isoclinism	isoclinic class of #93021
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1289
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×C₈×D₄	C₂²
prime radical	C₂²×D₄	Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

224 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fghi,d^2=h,e^2=fg,f^2=hi,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gi,c^b=ch,d^b=dg,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 218);