#93910

order	512
nilpotency class	5
isoclinism	isoclinic class of #93707
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1317
commutator	C₂³×C₄	C₂×C₈
Frattini, cosocle	#1728	C₂²
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1728 ◁ G
LECS	G ▷ #1728 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

227 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fi,d^2=g,g^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defi,g^b=gi,d^c=dh,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1106);