#6753279

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#37657	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6660475);