#102067

order	512
nilpotency class	3
isoclinism	isoclinic class of #9781
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

486 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=egh,d^2=gh,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9263);