#9007467

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	#55031	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3915 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)(3,8,5,6)(4,9,12,16)(11,15,14,13)(17,18,22,24)(19,23,27,32)(20,25,29,28)(21,30,31,26)(33,34)(35,43)(36,42)(37,41)(38,39)(40,44)(45,47)(46,48);(1,3,7,5)(2,8,10,6)(4,11,12,14)(9,15,16,13)(17,19,22,27)(18,23,24,32)(20,26,29,30)(21,28,31,25)(34,39)(35,44)(41,42)(45,46);(3,5)(4,12)(6,8)(9,16)(18,24)(19,27)(20,29)(21,31)(33,35)(34,40)(36,45)(37,46)(38,44)(39,43)(41,47)(42,48);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,28)(27,30)(31,32)(33,36,38,37)(34,41,39,42)(35,45,44,46)(40,47,43,48);(1,5,7,3)(2,6,10,8)(4,11,12,14)(9,15,16,13)(17,19)(18,23)(20,30)(21,28)(22,27)(24,32)(25,31)(26,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,44)(45,47)(46,48);(1,6)(2,3)(4,13)(5,10)(7,8)(9,11)(12,15)(14,16)(17,21,22,31)(18,26,24,30)(19,28,27,25)(20,23,29,32)(33,37,38,36)(34,42,39,41)(35,46,44,45)(40,48,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8914663);