#6803848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6803848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#39148
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1095 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)(3,14)(4,17)(5,23)(6,11)(7,28)(8,13)(9,21)(10,24)(12,27)(15,29)(16,19)(18,22)(20,26)(25,30)(31,32);(1,3)(2,4,13,22)(5,15)(6,26)(7,17)(8,9)(10,19,11,29)(12,14,28,21)(16,25)(18,27)(20,23,31,30)(24,32);(1,4)(2,9)(3,13)(5,19)(6,20)(7,14)(8,22)(10,16)(11,15)(12,18)(17,28)(21,27)(23,26)(24,31)(25,29)(30,32);(1,5,7,24,8,25,27,6)(2,10,12,23,13,11,28,30)(3,15,17,32,9,16,18,26)(4,19,14,31,22,29,21,20);(1,6,27,25,8,24,7,5)(2,11,12,30,13,10,28,23)(3,16,17,26,9,15,18,32)(4,20,21,29,22,31,14,19)

Polycyclic presentation:

2^9|b^2=fgh,d^2=f,e^2=fh,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,g^a=gi,h^a=hi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6711044);