#7678604

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

5132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585800);