#6993321

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38033	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fhi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

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, 6900517);