#8905174

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

4130 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8812370);