#7918203

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

4570 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,e^2=ghi,f^2=i,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, 7825399);