#10038471

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
prime radical	#44886	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4780 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,30,31,24)(20,26,32,25)(21,27,28,22);(1,3,5,6)(2,4,7,8)(9,10,12,14)(11,13,16,15)(17,19)(18,24)(20,22)(21,25)(23,31)(26,28)(27,32)(29,30);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,32)(18,25,29,26)(19,22,31,27)(21,30,28,24);(1,2)(3,8)(4,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,27)(21,24)(22,31)(23,32)(25,29)(28,30);(1,2)(3,4)(5,7)(6,8)(17,21)(18,27)(19,26)(20,24)(22,29)(23,28)(25,31)(30,32);(1,4)(2,6)(3,7)(5,8)(17,22,23,27)(18,28,29,21)(19,20,31,32)(24,26,30,25)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=i,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, 9945667);