#7850310

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4874 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7757506);