#7681760

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

4759 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)(5,7)(6,8)(9,10,13,12)(11,15,16,14)(21,29)(25,30)(26,31)(28,32);(1,3,6,5)(2,4,8,7)(9,11,13,16)(10,14,12,15)(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,30)(31,32);(1,3,6,5)(2,7,8,4)(18,23)(20,27)(25,30)(28,32);(1,4,6,7)(2,3,8,5)(17,19,22,24)(18,20,23,27)(21,26,29,31)(25,28,30,32);(1,3,6,5)(2,4,8,7)(9,12,13,10)(11,15,16,14)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32);(1,5,6,3)(2,7,8,4)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,28,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=hi,e^2=gh,f^2=hi,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, 7588956);