#9789211

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

5842 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326547, #72333953
maximal normal subgroup of	#72326547, #72333953
maximal quotient of	#72326604, #72326719, #72327476, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9696407);