#10324311

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

4993 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)(19,25)(20,22)(21,24)(23,27)(26,31)(28,29)(30,32);(1,2)(3,7)(4,5)(6,8)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(2,5)(6,7)(9,11,14,13)(10,15,12,16)(21,29)(24,28)(26,32)(30,31);(2,5)(6,7)(9,11,14,13)(10,16,12,15)(17,20,22,18)(19,23,27,25)(21,24,29,28)(26,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,16,13,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,16,12,15)(17,19)(18,25)(20,23)(21,26)(22,27)(24,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,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, 10231507);