#8634653

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

3945 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541849);