#8850573

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	#41122	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8757769);