#8811226

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	#49376	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=hi,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, 8718422);