#8695307

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:

(1,2,7,11)(3,12,14,8)(4,13,16,9)(5,6,10,15)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(34,39)(35,36)(42,45)(43,44);(1,3,7,14)(2,8,11,12)(4,15,16,6)(5,13,10,9)(18,23)(19,26)(25,28)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,4,7,16)(2,9,11,13)(3,6,14,15)(5,8,10,12)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,5)(2,6)(3,13)(4,8)(7,10)(9,14)(11,15)(12,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,35)(34,41,39,40)(37,43,46,44)(42,47,45,48);(1,6)(2,10)(3,4)(5,11)(7,15)(8,9)(12,13)(14,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(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=hi,d^2=gh,e^2=h,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, 8602503);