#10424688

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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)(3,11,12,7)(4,8,9,5)(13,14,16,15)(17,18,23,25)(19,21,27,20)(22,24,31,30)(26,29,32,28)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(3,12)(7,11)(13,16)(14,15)(17,19,23,27)(18,20,25,21)(22,26,31,32)(24,28,30,29)(35,39)(36,43)(42,47)(44,48);(1,3)(2,7)(4,13)(5,14)(6,12)(8,15)(9,16)(10,11)(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,30,32)(33,35,38,39)(34,36,41,43)(37,42,46,47)(40,44,45,48);(1,4)(2,8)(3,13)(5,10)(6,9)(7,15)(11,14)(12,16)(17,18)(19,20)(21,27)(22,30)(23,25)(24,31)(26,29)(28,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,48)(44,47);(1,3)(2,7)(4,13)(5,14)(6,12)(8,15)(9,16)(10,11)(17,21,23,20)(18,19,25,27)(22,29,31,28)(24,26,30,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,47)(42,45)(46,48);(1,5,6,8)(2,9,10,4)(3,14,12,15)(7,16,11,13)(17,22)(18,24)(19,26)(20,28)(21,29)(23,31)(25,30)(27,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,e^2=g,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, 10331884);