#93240

order	512
nilpotency class	5
isoclinism	isoclinic class of #93237
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1351
commutator	C₂×#95	C₂×C₄
Frattini, cosocle	#1445	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×#95 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ #1445 ◁ G
LECS	G ▷ #1445 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁

184 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=hi,e^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gh,c^b=cf,d^b=defgh,f^b=fi,g^b=gi,d^c=dgh,e^c=ei,e^d=eh

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 436);