#9070716

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

5870 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261968
maximal normal subgroup of	#72261968

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8977912);