#8622093

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	#41120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,e^2=gi,f^2=gh,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, 8529289);