#8818301

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4179 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,15)(14,16)(17,18,22,24)(19,23,27,32)(20,25,26,21)(28,31,30,29)(33,34,38,39)(35,41,43,36)(37,45,46,40)(42,44,48,47);(2,7)(5,14)(8,11)(9,15)(17,19)(18,23)(20,30)(21,31)(22,27)(24,32)(25,29)(26,28)(34,39)(36,41)(37,46)(42,48);(3,12)(4,13)(8,11)(9,15)(18,24)(19,27)(20,26)(29,31)(33,35,38,43)(34,36,39,41)(37,42,46,48)(40,44,45,47);(1,3,6,12)(2,8,7,11)(4,10,13,16)(5,9,14,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,32)(25,31)(26,30)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,4,6,13)(2,9,7,15)(3,10,12,16)(5,8,14,11)(17,20,22,26)(18,25,24,21)(19,28,27,30)(23,31,32,29)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(1,5,6,14)(2,10,7,16)(3,9,12,15)(4,8,13,11)(17,21,22,25)(18,26,24,20)(19,29,27,31)(23,30,32,28)(33,37,38,46)(34,40,39,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=ghi,e^2=gh,f^2=gi,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, 8725497);