#8838541

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

4021 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,e^2=g,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, 8745737);