#7881507

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

4941 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=gh,e^2=h,f^2=gi,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, 7788703);