#8202460

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=h,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, 8109656);