#8393359

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	#43194	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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)(3,7)(4,9)(5,13)(6,8)(10,16)(11,12)(14,15)(33,34)(35,42)(36,41)(37,46)(38,44)(39,40)(43,47)(45,48);(1,3,6,9)(2,7,8,4)(5,14,16,11)(10,12,13,15)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(34,40)(37,47)(41,42)(44,45);(2,8)(3,9)(10,13)(11,14)(18,23)(21,29)(24,25)(27,32)(33,35,39,36)(34,41,40,42)(37,44,47,45)(38,46,48,43);(1,4,6,7)(2,9,8,3)(5,15,16,12)(10,11,13,14)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,36,39,35)(34,42,40,41)(37,44,47,45)(38,46,48,43);(1,5,6,16)(2,10,8,13)(3,11,9,14)(4,12,7,15)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,37,39,47)(34,43,40,46)(35,44,36,45)(38,42,48,41);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,14)(13,16)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38,39,48)(34,44,40,45)(35,43,36,46)(37,42,47,41)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300555);