#9101910

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

5681 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261985, #72262220
maximal normal subgroup of	#72261985, #72262220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9009106);