#9634946

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
prime radical	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(9,10,13,15)(11,12,16,14)(17,18,23,20)(19,27,30,24)(21,25,33,31)(22,32,35,26)(28,39,41,36)(29,37,43,40)(34,44,45,38)(42,46,48,47);(1,2)(3,4)(5,7)(6,8)(9,11,13,16)(10,14,15,12)(21,33)(25,31)(28,41)(34,45)(36,39)(38,44)(42,48)(46,47);(1,3,5,8)(2,6,7,4)(9,11)(10,12)(13,16)(14,15)(17,19,23,30)(18,24,20,27)(21,28,33,41)(22,29,35,43)(25,36,31,39)(26,37,32,40)(34,42,45,48)(38,46,44,47);(1,3,5,8)(2,4,7,6)(9,12)(10,11)(13,14)(15,16)(17,20,23,18)(19,24,30,27)(21,31,33,25)(22,32,35,26)(28,36,41,39)(29,37,43,40)(34,44,45,38)(42,46,48,47);(1,3)(2,4)(5,8)(6,7)(17,21)(18,25)(19,28)(20,31)(22,34)(23,33)(24,36)(26,38)(27,39)(29,42)(30,41)(32,44)(35,45)(37,46)(40,47)(43,48);(1,4)(2,3)(5,6)(7,8)(17,22)(18,26)(19,29)(20,32)(21,34)(23,35)(24,37)(25,38)(27,40)(28,42)(30,43)(31,44)(33,45)(36,46)(39,47)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268346
Fitting subgroup of	#72268346
maximal subgroup of	#72268346
maximal normal subgroup of	#72268346
maximal quotient of	#72268312

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9542142);