#8968334

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

5398 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)(18,23)(19,25)(26,30)(27,32);(3,7)(6,8)(10,14)(11,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(19,25)(20,24)(21,29)(26,30);(10,14)(11,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)

Polycyclic presentation:

2^9|e^2=g,f^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, 8875530);