#10166662

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	#58071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34,39,43)(35,40,37,42)(36,46,38,41)(44,48,45,47);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35,39,37)(34,40,43,42)(36,45,38,44)(41,48,46,47);(2,4)(8,10)(9,12)(15,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36,39,38)(34,41,43,46)(35,44,37,45)(40,47,42,48);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,37,39,35)(34,42,43,40)(36,44,38,45)(41,47,46,48);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,38,39,36)(34,41,43,46)(35,45,37,44)(40,47,42,48);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,34)(35,40)(36,41)(37,42)(38,46)(39,43)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gi,e^2=gi,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, 10073858);