#8695844

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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=h,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, 8603040);