#9050395

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	#49743	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(4,11)(5,13)(6,9)(8,15)(10,12)(14,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3,6,11)(2,7,9,4)(5,10,15,16)(8,14,13,12)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(34,38)(36,44)(39,41)(42,48);(1,4,6,7)(2,3,9,11)(5,14,15,12)(8,16,13,10)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,2)(3,7)(4,11)(5,8)(6,9)(10,14)(12,16)(13,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,5,6,15)(2,8,9,13)(3,10,11,16)(4,12,7,14)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8957591);