#8602492

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	#41253	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=g,e^2=g,f^2=g,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, 8509688);