#9230858

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	#51968	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^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,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9138054);