#8622186

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	#58212	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(18,23)(19,26)(20,28)(30,31)(33,34)(35,40)(36,45)(37,44)(38,41)(39,42)(43,48)(46,47);(1,3,7,8)(2,5,10,13)(4,11,9,14)(6,12,15,16)(17,18)(19,21)(20,29)(22,23)(24,28)(25,26)(27,30)(31,32)(33,35)(34,40)(36,46)(37,39)(38,43)(41,48)(42,44)(45,47);(3,8)(5,13)(11,14)(12,16)(17,19,22,26)(18,21,23,25)(20,27,28,32)(24,31,29,30)(33,36,39,38)(34,41,42,45)(35,43,37,46)(40,47,44,48);(1,4)(2,6)(3,11)(5,12)(7,9)(8,14)(10,15)(13,16)(17,20)(18,24)(19,27)(21,31)(22,28)(23,29)(25,30)(26,32)(33,34)(35,44)(36,41)(37,40)(38,45)(39,42)(43,48)(46,47);(1,5,7,13)(2,8,10,3)(4,12,9,16)(6,14,15,11)(17,19)(18,25)(20,27)(21,23)(22,26)(24,30)(28,32)(29,31)(33,37,39,35)(34,40,42,44)(36,43,38,46)(41,48,45,47);(1,6)(2,9)(3,12)(4,10)(5,14)(7,15)(8,16)(11,13)(17,21,22,25)(18,19,23,26)(20,30,28,31)(24,32,29,27)(33,38,39,36)(34,41,42,45)(35,43,37,46)(40,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,e^2=i,f^2=h,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, 8529382);