#9007157

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	#35733	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,23,26)(19,24,21,22)(20,25,28,32)(27,31,30,29)(33,34,38,40)(35,41,43,37)(36,39,46,44)(42,47,48,45);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,23,21)(18,24,26,22)(20,27,28,30)(25,31,32,29)(33,35)(34,37)(36,42)(38,43)(39,45)(40,41)(44,47)(46,48);(2,6)(3,5)(4,12)(15,16)(19,21)(20,28)(22,24)(25,32)(33,34,38,40)(35,37,43,41)(36,44,46,39)(42,47,48,45);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,34)(35,37)(36,44)(38,40)(39,46)(41,43)(42,47)(45,48);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,21,23,19)(18,22,26,24)(20,27,28,30)(25,31,32,29)(33,36,38,46)(34,39,40,44)(35,42,43,48)(37,45,41,47);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22)(18,19)(20,29)(21,26)(23,24)(25,27)(28,31)(30,32)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8914353);