#10481030

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

4802 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388226);