#6371103

order	512
nilpotency class	3
isoclinism	isoclinic class of #6370843
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#117
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 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)(3,12,15,8)(4,9)(5,10)(6,11)(7,14,20,28)(13,25,26,21)(16,22)(17,23)(18,24)(19,27,30,32)(29,31);(1,3)(2,7,11,20)(4,13)(5,14)(6,15)(8,10,12,24)(9,19,22,30)(16,26)(17,27)(18,28)(21,23,25,31)(29,32);(1,3)(2,8)(4,13)(5,14)(6,15)(7,10)(9,21)(11,12)(16,26)(17,27)(18,28)(19,23)(20,24)(22,25)(29,32)(30,31);(1,4,5,17,6,16,18,29)(2,9,10,23,11,22,24,31)(3,13,14,27,15,26,28,32)(7,19,12,25,20,30,8,21);(4,16)(9,22)(13,26)(17,29)(19,30)(21,25)(23,31)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278299);