#8614114

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

5377 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,e^2=i,f^2=gh,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	#72304012
maximal normal subgroup of	#72304012
maximal quotient of	#72304136

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8521310);