#8746940

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3813 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8654136);