#369702

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1236 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=eg,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di

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, 276898);