#380264

order	512
nilpotency class	3
isoclinism	isoclinic class of #380263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14262
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287460);