#9005189

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4065 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8912385);