#8837062

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

4094 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)(6,14)(7,8)(10,15)(11,16)(12,13)(17,18)(19,25)(20,27)(21,29)(22,23)(24,31)(26,28)(30,32)(33,34,38,39)(35,40,46,41)(36,37,43,42)(44,45,47,48);(1,3,7,4)(2,5,8,9)(6,12,15,11)(10,16,14,13)(17,19,22,27)(18,20,23,25)(21,30,31,26)(24,32,29,28)(34,39)(37,42)(40,41)(45,48);(2,8)(3,4)(10,14)(11,12)(18,23)(20,25)(24,29)(28,32)(33,35,38,46)(34,40,39,41)(36,44,43,47)(37,45,42,48);(1,4,7,3)(2,9,8,5)(6,12,15,11)(10,16,14,13)(17,20,22,25)(18,19,23,27)(21,28,31,32)(24,26,29,30)(33,35)(34,41)(36,47)(37,45)(38,46)(39,40)(42,48)(43,44);(1,5,7,9)(2,4,8,3)(6,13,15,16)(10,12,14,11)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36,38,43)(34,42,39,37)(35,44,46,47)(40,48,41,45);(1,6,7,15)(2,10,8,14)(3,11,4,12)(5,13,9,16)(17,21)(18,24)(19,26)(20,28)(22,31)(23,29)(25,32)(27,30)(33,37,38,42)(34,43,39,36)(35,45,46,48)(40,47,41,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8744258);