#9545641

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

4158 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(2,7)(4,14)(8,11)(10,15)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(2,7)(3,12)(4,14)(5,16)(18,25)(19,20)(26,32)(27,28)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9452837);