#7903876

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

4675 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:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=i,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7811072);