#10159198

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	#55583	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10066394);