#8551138

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

4248 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)(3,10,8,4)(5,9,14,15)(11,16,13,12)(17,18,22,23)(19,20,27,25)(21,24,30,31)(26,28,29,32)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(2,7)(3,8)(9,15)(11,13)(17,18,22,23)(19,25,27,20)(21,24,30,31)(26,32,29,28)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(2,7)(4,10)(9,15)(12,16)(18,23)(20,25)(24,31)(28,32)(37,45)(40,47)(42,48)(44,46);(1,3,6,8)(2,4,7,10)(5,13,14,11)(9,16,15,12)(17,19)(18,20)(21,29)(22,27)(23,25)(24,32)(26,30)(28,31)(33,36,38,39)(34,35,41,43)(37,44,45,46)(40,42,47,48);(1,4)(2,8)(3,7)(5,12)(6,10)(9,13)(11,15)(14,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,5,6,14)(2,9,7,15)(3,11,8,13)(4,12,10,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=g,f^2=hi,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, 8458334);