#6374624

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342674
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3490	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,h^c=hi,e^d=ei

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, 6281820);