#8291668

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	#55640	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3924 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198864);