#8968323

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

5175 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279724
maximal normal subgroup of	#72279724

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8875519);