#8982113

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

5002 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,24)(19,25,26,21)(20,23,28,31)(27,32,29,30);(3,8)(6,7)(9,10,12,14)(11,13,16,15)(17,18)(19,21)(20,23)(22,24)(25,26)(27,30)(28,31)(29,32);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,14)(15,16)(19,26)(20,28)(21,25)(23,31);(3,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,29)(22,26)(23,32)(24,25)(27,28)(30,31);(1,3)(2,7)(4,6)(5,8)(17,20)(18,23)(19,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,4,5,2)(3,6,8,7)(17,21)(18,19)(20,30)(22,25)(23,27)(24,26)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,f^2=h,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	#72303333
maximal normal subgroup of	#72303333
maximal quotient of	#72303430

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8889309);