#9390493

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	#48471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297689);