#7685310

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

5439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592506);