#8587878

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(3,10)(4,14)(5,9)(6,8)(7,13)(11,16)(12,15)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,34)(35,40)(36,37)(38,43)(39,41)(42,47)(44,48)(45,46);(1,3,6,13)(2,7,8,10)(4,15,9,11)(5,12,14,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32)(33,35,39,44)(34,40,41,48)(36,45,42,43)(37,46,47,38);(2,8)(4,9)(7,10)(11,15)(21,30)(24,32)(26,29)(28,31)(34,41)(35,44)(37,47)(43,45);(1,3)(2,7)(4,11)(5,16)(6,13)(8,10)(9,15)(12,14)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,36,39,42)(34,37,41,47)(35,43,44,45)(38,48,46,40);(1,4,6,9)(2,5,8,14)(3,11,13,15)(7,16,10,12)(17,21,22,30)(18,24,25,32)(19,26,27,29)(20,28,23,31)(33,37)(34,42)(35,38)(36,41)(39,47)(40,45)(43,48)(44,46);(1,5)(2,9)(3,12)(4,8)(6,14)(7,11)(10,15)(13,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,38)(34,43)(35,37)(36,40)(39,46)(41,45)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=g,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, 8495074);