#9054475

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	#51084	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=g,e^2=gi,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, 8961671);