#10196261

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	#50760	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=h,e^2=g,f^2=gi,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, 10103457);