#10440012

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	#59689	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=g,e^2=i,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, 10347208);