#8664380

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

4111 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gh,e^2=h,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, 8571576);