#10446490

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10353686);