#8617787

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

4030 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8524983);