#8691000

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	#37903	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8598196);