#10060934

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

4967 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)(9,10)(11,15)(12,13)(14,16)(17,18,23,26)(19,24,22,32)(20,25,30,21)(27,28,29,31);(1,3,5,8)(2,6,7,4)(10,13)(14,15)(17,19,23,22)(18,24,26,32)(20,27,30,29)(21,31,25,28);(2,7)(3,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,30)(18,21,26,25)(19,27,22,29)(24,31,32,28);(1,3)(2,4)(5,8)(6,7)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,27)(21,26)(22,29)(23,30)(24,28)(31,32);(1,3)(2,6)(4,7)(5,8)(17,21,23,25)(18,20,26,30)(19,28,22,31)(24,29,32,27);(1,4,5,6)(2,8,7,3)(17,22,23,19)(18,24,26,32)(20,29,30,27)(21,31,25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,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, 9968130);