#10216951

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	#54330	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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=hi,e^2=gi,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, 10124147);