#8407012

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

4374 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)(3,8)(4,13)(5,9)(6,7)(10,11)(12,15)(14,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(35,36)(37,45)(39,42)(40,46);(2,7)(4,9)(8,15)(10,14)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(33,34)(35,42)(36,39)(37,46)(38,41)(40,45)(43,47)(44,48);(2,7)(5,13)(8,15)(11,16)(19,28)(22,31)(24,32)(26,29)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,47,46,48);(1,3,6,12)(2,8,7,15)(4,14,9,10)(5,11,13,16)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,36,38,35)(34,39,41,42)(37,43,45,44)(40,48,46,47);(1,4,6,9)(2,5,7,13)(3,10,12,14)(8,16,15,11)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,37)(34,40)(35,43)(36,44)(38,45)(39,47)(41,46)(42,48);(1,5)(2,9)(3,11)(4,7)(6,13)(8,10)(12,16)(14,15)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314208);