#10368586

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

5106 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)(11,12)(13,16)(14,15)(17,18,22,24)(19,25,27,21)(20,23,28,31)(26,32,29,30);(1,2,5,7)(3,4,8,6)(10,15)(12,13)(17,19,22,27)(18,21,24,25)(20,26,28,29)(23,30,31,32);(1,3,5,8)(2,6,7,4)(9,11,14,16)(10,13,15,12)(20,28)(23,31)(26,29)(30,32);(1,3,5,8)(2,6,7,4)(9,12,14,13)(10,16,15,11)(17,19)(18,21)(20,29)(22,27)(23,32)(24,25)(26,28)(30,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,20,22,28)(18,23,24,31)(19,26,27,29)(21,30,25,32);(1,4)(2,3)(5,6)(7,8)(9,13,14,12)(10,11,15,16)(17,21)(18,19)(20,30)(22,25)(23,26)(24,27)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=hi,e^2=gh,f^2=h,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, 10275782);