#8650346

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

5304 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)(3,5)(4,7)(6,8)(10,13)(14,16)(17,18,22,24)(19,26,21,23)(20,28,30,25)(27,31,29,32);(2,3)(6,7)(11,15)(14,16)(17,19,22,21)(18,23,24,26)(20,29,30,27)(25,32,28,31);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,24)(23,26)(25,28)(31,32);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,28)(26,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,30)(29,31);(17,21,22,19)(18,23,24,26)(20,27,30,29)(25,32,28,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304009
maximal normal subgroup of	#72304009
maximal quotient of	#72304120

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557542);