#10451632

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3866 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=hi,e^2=gi,f^2=ghi,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, 10358828);