#9058955

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

4145 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)(3,11,12,8)(4,9,14,13)(5,15,16,10)(33,34,37,41)(35,39,42,38)(36,40,45,48)(43,47,44,46);(2,7)(3,12)(9,13)(10,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(35,42)(38,39)(43,44)(46,47);(2,7)(3,12)(4,14)(5,16)(18,22)(19,26)(20,29)(31,32)(33,35,37,42)(34,38,41,39)(36,44,45,43)(40,47,48,46);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,39)(36,44)(37,42)(38,41)(40,46)(43,45)(47,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36,37,45)(34,40,41,48)(35,43,42,44)(38,46,39,47);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,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,d^c=dgi,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, 8966151);