#10368057

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

4740 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=gi,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, 10275253);