#92826

order	512
nilpotency class	4
isoclinism	isoclinic class of #92825
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1270
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	C₄×#95	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₄×#95 ◁ G
LECS	G ▷ C₄×#95 ▷ C₂³×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 276 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227432
maximal normal subgroup of	#72227432
maximal quotient of	#72268637

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22);