#8534607

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

4262 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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(17,18)(19,24)(20,26)(21,30)(22,27)(23,25)(28,32)(29,31)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(2,7)(3,9)(10,15)(11,13)(17,19,23,21)(18,24,25,30)(20,28,29,27)(22,26,32,31)(34,35)(39,41)(40,42)(46,47);(2,7)(4,8)(10,15)(12,16)(18,25)(19,21)(26,31)(27,28)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,20,23,29)(18,26,25,31)(19,27,21,28)(22,30,32,24)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,21,23,19)(18,24,25,30)(20,28,29,27)(22,31,32,26)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5,6,14)(2,10,7,15)(3,11,9,13)(4,12,8,16)(17,22)(18,27)(19,26)(20,24)(21,31)(23,32)(25,28)(29,30)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

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

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, 8441803);