#6803854

order	512
nilpotency class	3
isoclinism	isoclinic class of #6803853
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

1099 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:

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

Polycyclic presentation:

2^9|a^2=i,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,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6711050);