#9490136

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	#55224	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(33,34)(35,43)(36,41)(37,42)(38,39)(40,44)(45,48)(46,47);(2,6)(7,9)(8,12)(14,15)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(35,44)(37,45)(40,43)(42,48);(2,6)(3,10)(4,13)(14,15)(18,23)(20,27)(21,29)(26,31)(33,35,39,44)(34,40,38,43)(36,37,47,45)(41,48,46,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32)(33,36,39,47)(34,41,38,46)(35,45,44,37)(40,42,43,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,37,39,45)(34,42,38,48)(35,36,44,47)(40,46,43,41);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38,39,34)(35,40,44,43)(36,46,47,41)(37,48,45,42)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9397332);