#9910951

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	#49857	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9818147);