#10457948

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

5146 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)(10,12)(14,15)(17,18,22,24)(19,23,28,32)(20,21,29,31)(25,26,27,30);(9,10,13,12)(11,15,16,14)(17,19)(18,23)(20,27)(21,30)(22,28)(24,32)(25,29)(26,31);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,20)(18,21)(19,25)(22,29)(23,26)(24,31)(27,28)(30,32);(3,7)(6,8)(9,12,13,10)(11,15,16,14)(17,21)(18,20)(19,26)(22,31)(23,25)(24,29)(27,32)(28,30);(1,3)(2,6)(4,8)(5,7)(17,20)(18,21)(19,27)(22,29)(23,30)(24,31)(25,28)(26,32);(1,4,5,2)(3,8,7,6)(17,19)(18,23)(20,25)(21,26)(22,28)(24,32)(27,29)(30,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,f^2=h,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, 10365144);