#93716

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1831	C₂²
socle	C₂²	#1319
prime radical	C₂³	#323
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1831 ◁ G
LECS	G ▷ #1831 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

138 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,4,9,5,7,13,14,6,8,12,15,3,10,11,16)(17,18)(19,27,29,42)(20,31,33,47)(21,35,37,25)(22,26)(23,28,38,45)(24,34)(30,44,36,43)(32,39,46,40)(41,48);(1,3,6,5)(2,7,8,10)(4,11,12,13)(9,16,15,14)(17,19)(18,23,26,40)(20,32)(21,36)(22,30)(24,27,41,44)(25,38,35,39)(28,33)(29,37)(31,42,47,43)(34,46)(45,48)

Polycyclic presentation:

2^9|a^2=d,b^2=g,c^2=gh,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cghi,d^b=deghi,e^b=ei,d^c=dgi,e^d=ei

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, 912);