#8618128

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

4160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8525324);