#7931622

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

4610 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=ghi,e^2=h,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, 7838818);