#10320956

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

4937 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)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32);(1,2)(3,7)(4,5)(6,8)(17,19,22,30)(18,23,26,27)(20,21,29,28)(24,25,31,32);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31);(2,5)(6,7)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,21)(22,29)(23,25)(26,31)(27,32)(28,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,25)(19,29)(20,30)(22,28)(23,31)(24,27)(26,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^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, 10228152);