#8843581

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

4180 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=ghi,e^2=gh,f^2=h,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, 8750777);