#9002445

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	#51131	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,e^2=hi,f^2=h,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, 8909641);