#9040808

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

4123 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948004);