#102034

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

400 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,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, 9230);