#7877209

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

4721 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7784405);