#8664438

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

4092 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8571634);