#8039011

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

4590 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)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32);(1,3,6,7)(2,4,5,8)(9,11,13,16)(10,14,12,15)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(1,2,6,5)(3,8,7,4)(9,10,13,12)(11,14,16,15)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31);(1,4)(2,7)(3,5)(6,8)(9,10,13,12)(11,14,16,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,4)(2,3)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32);(1,5,6,2)(3,8,7,4)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7946207);