#8202227

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	#53698	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,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,f^c=fgi

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, 8109423);