#10440024

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	#48106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4105 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)(3,4,11,8)(5,9,14,15)(10,12,13,16)(17,18)(19,23)(20,25)(21,31)(22,27)(24,30)(26,28)(29,32)(34,40)(35,36)(38,42)(44,46);(2,7)(4,8)(9,15)(12,16)(17,19,22,29)(18,23,27,32)(20,28,30,21)(24,31,25,26)(33,34,39,40)(35,43,36,41)(37,42,48,38)(44,45,46,47);(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,27)(26,32)(33,35,39,36)(34,41,40,43)(37,44,48,46)(38,45,42,47);(1,3,6,11)(2,8,7,4)(5,13,14,10)(9,12,15,16)(17,20)(18,25)(19,28)(21,29)(22,30)(23,26)(24,27)(31,32)(33,36,39,35)(34,41,40,43)(37,44,48,46)(38,47,42,45);(1,4,6,8)(2,3,7,11)(5,12,14,16)(9,10,15,13)(17,18)(19,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,37)(34,42)(35,44)(36,46)(38,40)(39,48)(41,47)(43,45);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,21,22,28)(18,26,27,31)(19,20,29,30)(23,24,32,25)(33,38,39,42)(34,37,40,48)(35,45,36,47)(41,44,43,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=g,e^2=gi,f^2=h,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, 10347220);