#8324249

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

4235 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)(3,8)(4,6)(5,9)(7,11)(10,15)(12,13)(14,16)(17,18,23,24)(19,27,28,22)(20,29,32,25)(21,31,30,26)(33,34)(35,36)(37,45)(38,39)(40,47)(41,44)(42,46)(43,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,19,23,28)(18,22,24,27)(20,30,32,21)(25,26,29,31)(33,35,38,41)(34,36,39,44)(37,42,47,48)(40,43,45,46);(5,13)(9,12)(10,16)(14,15)(18,24)(22,27)(25,29)(26,31)(34,39)(35,41)(40,45)(42,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20,23,32)(18,25,24,29)(19,21,28,30)(22,31,27,26)(33,36)(34,35)(37,46)(38,44)(39,41)(40,48)(42,45)(43,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21)(18,26)(19,20)(22,29)(23,30)(24,31)(25,27)(28,32)(33,37)(34,40)(35,42)(36,43)(38,47)(39,45)(41,48)(44,46);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,22,23,27)(18,19,24,28)(20,31,32,26)(21,29,30,25)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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, 8231445);