#9061101

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

5421 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)(11,16)(12,14)(17,18,23,28)(19,29,32,24)(20,22,27,25)(21,26,30,31);(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,15)(14,16)(17,19,23,32)(18,24,28,29)(20,21,27,30)(22,31,25,26);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(20,27)(21,30)(22,25)(26,31);(10,15)(11,16)(17,20,23,27)(18,25,28,22)(19,30,32,21)(24,31,29,26);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,21,23,30)(18,26,28,31)(19,20,32,27)(22,24,25,29);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,22,23,25)(18,27,28,20)(19,31,32,26)(21,24,30,29)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279754
maximal normal subgroup of	#72279754
maximal quotient of	#72280568, #72280669, #72281005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968297);