#8324160

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

4242 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231356);