#9058394

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

4118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gh,e^2=g,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, 8965590);