#99860

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#386
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

438 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=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, 7056);