#9243459

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,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, 9150655);