#8334603

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

4060 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=hi,e^2=gh,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, 8241799);