#93019

order	512
nilpotency class	4
isoclinism	isoclinic class of #93011
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

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

228 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,6,8,5)(4,7)(9,10,13,18,15,21,27,20)(11,22,24,36,26,37,38,31)(12,19,14,17,28,34,29,35)(16,25,30,39,33,40,32,23);(2,5,7,6)(9,11,14,25,13,24,28,39,15,26,29,40,27,38,12,23)(10,16,19,31,20,32,35,37,21,33,34,36,18,30,17,22)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fghi,d^2=hi,e^2=fgi,f^2=hi,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,d^b=dg,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 215);