#9240977

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	#55640	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9148173);