#8599667

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

5743 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)(4,6)(7,8)(17,18,24,28)(19,29,30,25)(20,27,21,26)(22,31,32,23);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16)(17,19,24,30)(18,25,28,29)(20,23,21,31)(22,27,32,26);(2,4)(3,7)(9,10,12,13)(11,14,15,16)(17,20,24,21)(18,26,28,27)(19,23,30,31)(22,29,32,25);(3,7)(5,8)(11,15)(14,16)(17,21,24,20)(18,27,28,26)(19,23,30,31)(22,29,32,25);(1,4,6,2)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,23)(19,26)(20,25)(21,29)(24,32)(27,30)(28,31);(1,5)(2,3)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,23)(18,22)(19,21)(20,30)(24,31)(25,27)(26,29)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=g,e^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	#72304020
maximal normal subgroup of	#72304020
maximal quotient of	#72304061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506863);