#10060932

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

5124 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9968128);