#7643387

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550583);