#9057767

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

5586 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)(9,10,13,15)(11,12,16,14)(17,18)(19,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32);(1,2,4,6)(3,5,7,8)(9,11,13,16)(10,14,15,12)(17,19,22,27)(18,23,25,20)(21,26,31,32)(24,30,29,28);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26);(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,31)(18,24,25,29)(19,26,27,32)(20,28,23,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261985, #72262112
maximal normal subgroup of	#72261985, #72262112

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8964963);