#8664387

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

4202 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,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, 8571583);