#9040304

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	#49795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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)(3,10,11,8)(4,5,9,15)(12,13,14,16)(34,39)(35,36)(41,46)(42,44);(2,7)(3,11)(4,9)(13,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,34)(35,40)(36,43)(37,41)(38,39)(42,47)(44,48)(45,46);(4,9)(5,15)(12,14)(13,16)(18,22)(20,29)(23,25)(26,30)(33,35,38,36)(34,40,39,43)(37,44,45,42)(41,48,46,47);(1,3,6,11)(2,8,7,10)(4,14,9,12)(5,13,15,16)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,36,38,35)(34,40,39,43)(37,42,45,44)(41,48,46,47);(1,4,6,9)(2,5,7,15)(3,12,11,14)(8,16,10,13)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,37,38,45)(34,41,39,46)(35,42,36,44)(40,47,43,48);(1,5)(2,9)(3,13)(4,7)(6,15)(8,12)(10,14)(11,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,34)(35,43)(36,40)(37,41)(38,39)(42,48)(44,47)(45,46)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947500);