#10368508

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

4920 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275704);