#8639735

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	#40054	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,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, 8546931);