#6993250

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	#38909	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1633 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)(17,18,23,27,22,26,21,28)(19,20,30,31,25,24,29,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,22,25)(18,24,26,20)(21,29,23,30)(27,32,28,31);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,20)(18,25)(19,26)(21,31)(22,24)(23,32)(27,30)(28,29);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(18,26)(19,25)(27,28)(29,30);(9,13,14,12)(10,15,11,16)(17,21,22,23)(18,27,26,28)(19,29,25,30)(20,31,24,32)

Polycyclic presentation:

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

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, 6900446);