#243359

order	512
nilpotency class	3
isoclinism	isoclinic class of #233055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150555);