#8843928

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

4224 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=hi,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, 8751124);