#93220

order	512
nilpotency class	5
isoclinism	isoclinic class of #93213
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
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

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

normal subgroups only

This group has 217 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)(4,8,14,7)(5,9)(12,16,13,15)(17,18,20,25,22,27,34,44)(19,28,29,42,31,39,46,47)(21,26,36,43,37,24,33,41)(23,32,38,35,40,45,48,30);(1,3,5,12)(2,6,9,15)(4,13,14,11)(7,10,8,16)(17,19,22,31)(18,23,27,40)(20,32,34,45)(21,35,37,30)(24,39,26,28)(25,42,44,47)(29,33,46,36)(38,43,48,41)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267851
maximal normal subgroup of	#72267851
maximal quotient of	#72267854

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416);