#10328401

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

4963 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,e^2=g,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, 10235597);