#8773016

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	#36811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(2,6)(4,13)(7,9)(11,16)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(34,40)(35,44)(37,47)(42,45);(2,6)(3,10)(8,12)(11,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,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, 8680212);