#7684858

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

4858 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=g,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, 7592054);