#9058273

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
prime radical	#46393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4164 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)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(33,34)(35,39)(36,40)(37,43)(38,41)(42,47)(44,46)(45,48);(2,7)(3,9)(4,12)(14,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35,38,43)(34,39,41,37)(36,42,46,48)(40,47,44,45);(3,9)(4,12)(5,10)(8,15)(18,22)(19,26)(20,29)(31,32)(33,34,38,41)(35,37,43,39)(36,44,46,40)(42,47,48,45);(1,3,6,9)(2,5,7,10)(4,13,12,11)(8,16,15,14)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,34)(35,37)(36,44)(38,41)(39,43)(40,46)(42,47)(45,48);(1,4,6,12)(2,8,7,15)(3,11,9,13)(5,14,10,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48);(1,5)(2,9)(3,7)(4,14)(6,10)(8,13)(11,15)(12,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=g,e^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, 8965469);