#6757310

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	#37846	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1812 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)(9,10)(11,14)(12,13)(15,16)(17,18,23,32,22,26,21,25)(19,20,27,29,24,31,28,30);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,22,24)(18,20,26,31)(21,28,23,27)(25,30,32,29);(9,11,12,15)(10,14,13,16)(17,20,22,31)(18,24,26,19)(21,29,23,30)(25,28,32,27);(2,6)(5,7)(10,13)(14,16)(17,21,22,23)(18,25,26,32)(19,27,24,28)(20,29,31,30);(3,8)(5,7)(18,26)(19,24)(25,32)(27,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,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, 6664506);