#483435

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ch,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260401, #72260406, #72260411
maximal normal subgroup of	#72260401, #72260406, #72260411

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390631);