#8928038

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

3979 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)(4,7)(5,13)(6,8)(9,14)(11,15)(12,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(34,39)(37,45)(40,46)(44,48);(1,3,6,4)(2,7,8,10)(5,11,14,12)(9,15,13,16)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(36,43)(40,46)(42,47)(44,48);(2,8)(5,14)(7,10)(11,12)(19,27)(20,28)(22,32)(24,29)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(1,2)(3,7)(4,10)(5,9)(6,8)(11,15)(12,16)(13,14)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,35,38,41)(34,37,39,45)(36,42,43,47)(40,44,46,48);(1,4,6,3)(2,7,8,10)(5,12,14,11)(9,15,13,16)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,36,38,43)(34,40,39,46)(35,42,41,47)(37,44,45,48);(1,5,6,14)(2,9,8,13)(3,11,4,12)(7,15,10,16)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835234);