#8202020

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

4240 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^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, 8109216);