#9240902

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	#54856	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,e^2=h,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, 9148098);