#8701519

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=ghi,e^2=gh,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, 8608715);