#10324075

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

4742 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,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, 10231271);