#9236502

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

5397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327258
maximal normal subgroup of	#72327258
maximal quotient of	#72327299, #72327692, #72328131

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9143698);