#8692576

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	#42033	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=h,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, 8599772);