#8617785

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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:

(1,2,5,6)(3,9,10,7)(4,8,13,14)(11,16,12,15)(17,18,23,27)(19,24,21,25)(20,22,26,32)(28,29,30,31)(33,34)(35,39)(36,46)(37,38)(40,41)(42,47)(43,44)(45,48);(2,6)(3,10)(8,14)(11,12)(17,19)(18,24)(20,28)(21,23)(22,29)(25,27)(26,30)(31,32)(33,35,40,45)(34,39,41,48)(36,43,37,42)(38,47,46,44);(2,6)(7,9)(8,14)(15,16)(19,21)(24,25)(28,30)(29,31)(33,36,40,37)(34,38,41,46)(35,42,45,43)(39,44,48,47);(1,3,5,10)(2,7,6,9)(4,12,13,11)(8,16,14,15)(17,20)(18,22)(19,28)(21,30)(23,26)(24,29)(25,31)(27,32)(33,37,40,36)(34,38,41,46)(35,43,45,42)(39,44,48,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21,23,19)(18,25,27,24)(20,30,26,28)(22,31,32,29)(33,38)(34,36)(35,44)(37,41)(39,42)(40,46)(43,48)(45,47);(1,4,5,13)(2,8,6,14)(3,11,10,12)(7,15,9,16)(17,22)(18,26)(19,29)(20,27)(21,31)(23,32)(24,30)(25,28)(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=gh,c^2=gh,d^2=h,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, 8524981);