#8806912

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	#46707	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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)(3,6)(4,11)(5,7)(8,13)(9,15)(10,16)(12,14)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34)(35,42)(36,46)(37,41)(38,39)(40,45)(43,44)(47,48);(1,3,5,9)(2,6,7,15)(4,12,13,10)(8,16,11,14)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(34,38)(37,46)(40,42)(44,48);(2,7)(3,9)(8,11)(10,12)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,40)(36,47)(37,48)(38,42)(39,45)(41,43)(44,46);(1,2)(3,6)(4,8)(5,7)(9,15)(10,14)(11,13)(12,16)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,36)(34,37)(35,43)(38,46)(39,41)(40,44)(42,48)(45,47);(1,3)(2,6)(4,10)(5,9)(7,15)(8,14)(11,16)(12,13)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,37,39,46)(34,41,38,36)(35,44,45,48)(40,47,42,43);(1,4,5,13)(2,8,7,11)(3,10,9,12)(6,14,15,16)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38,39,34)(35,42,45,40)(36,46,41,37)(43,48,47,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=ghi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714108);