#8695869

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=gh,e^2=g,f^2=g,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, 8603065);