#8813995

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

4189 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)(3,5)(4,9)(6,7)(8,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,26)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,40)(36,37)(38,39)(41,44)(42,48)(43,45)(46,47);(2,7)(4,12)(5,8)(11,16)(17,19,23,20)(18,24,25,26)(21,30,27,28)(22,31,32,29)(34,39)(37,47)(40,41)(43,48);(2,7)(3,10)(4,12)(13,15)(18,25)(19,20)(21,27)(29,31)(33,35,38,44)(34,40,39,41)(36,45,46,42)(37,43,47,48);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,20,23,19)(18,26,25,24)(21,30,27,28)(22,31,32,29)(33,35)(34,41)(36,42)(37,43)(38,44)(39,40)(45,46)(47,48);(1,4,6,12)(2,9,7,14)(3,11,10,16)(5,13,8,15)(17,21,23,27)(18,22,25,32)(19,28,20,30)(24,29,26,31)(33,36)(34,37)(35,42)(38,46)(39,47)(40,48)(41,43)(44,45);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,22)(18,27)(19,29)(20,31)(21,25)(23,32)(24,30)(26,28)(33,37)(34,36)(35,43)(38,47)(39,46)(40,45)(41,42)(44,48)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=h,e^2=h,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, 8721191);