#8634448

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	#35109	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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)(17,18,23,27)(19,24,30,29)(20,31,21,25)(22,26,32,28)(33,34)(35,42)(36,46)(37,45)(38,40)(39,41)(43,47)(44,48);(1,3)(2,8)(4,11)(5,6)(7,10)(9,15)(12,14)(13,16)(17,19,23,30)(18,24,27,29)(20,28,21,26)(22,25,32,31)(33,35,40,45)(34,37,38,42)(36,43,39,44)(41,47,46,48);(3,10)(5,8)(11,16)(12,15)(17,20,23,21)(18,25,27,31)(19,26,30,28)(22,29,32,24)(33,36,40,39)(34,41,38,46)(35,43,45,44)(37,47,42,48);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,21,23,20)(18,25,27,31)(19,28,30,26)(22,29,32,24)(33,37)(34,35)(36,47)(38,45)(39,48)(40,42)(41,43)(44,46);(1,5)(2,3)(4,12)(6,10)(7,8)(9,11)(13,15)(14,16)(17,18)(19,29)(20,31)(21,25)(22,26)(23,27)(24,30)(28,32)(33,38,40,34)(35,37,45,42)(36,41,39,46)(43,48,44,47);(1,6,7,2)(3,5,10,8)(4,9,13,14)(11,15,16,12)(17,22,23,32)(18,26,27,28)(19,25,30,31)(20,29,21,24)(33,39,40,36)(34,41,38,46)(35,44,45,43)(37,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gh,e^2=g,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, 8541644);