#9059096

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	#43155	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(34,39)(35,44)(37,42)(45,47);(1,3)(2,8)(4,7)(5,12)(6,13)(9,10)(11,16)(14,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,34)(35,40)(36,42)(37,43)(38,39)(41,44)(45,48)(46,47);(2,9)(3,4)(5,15)(6,16)(18,22)(19,26)(20,29)(31,32)(33,35,38,44)(34,40,39,41)(36,47,43,45)(37,48,42,46);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,41)(36,47)(37,46)(38,44)(39,40)(42,48)(43,45);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36,38,43)(34,42,39,37)(35,45,44,47)(40,48,41,46);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,42)(34,43,39,36)(35,46,44,48)(40,47,41,45)

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=h,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, 8966292);