#8919224

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

4181 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8826420);