#8366619

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

4047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8273815);