#102016

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#390
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 408 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) = 512

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

(2,6,7,8)(9,10,12,14,13,16,11,15)(17,18)(19,24)(20,29)(21,31)(22,26)(23,27)(25,30)(28,32);(1,2,4,8,5,7,3,6)(10,14)(11,12)(15,16)(17,19,21,27,22,28,30,20)(18,23,25,24,26,29,31,32);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,15,16,14)(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=df,c^2=egh,d^2=gh,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ci,d^b=dh,f^b=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9212);