#7919028

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

4705 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=g,e^2=gi,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, 7826224);