#8700383

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607579);