#7696533

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

4589 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603729);