#8664419

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

4218 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18)(19,27)(20,25)(21,30)(22,24)(23,26)(28,32)(29,31)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(2,7)(3,10)(8,14)(11,16)(17,19,23,22)(18,24,26,27)(20,28,21,29)(25,31,30,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,20,23,21)(18,25,26,30)(19,28,22,29)(24,31,27,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,21,23,20)(18,25,26,30)(19,29,22,28)(24,31,27,32)(33,34)(35,43)(36,40)(37,48)(38,42)(39,46)(41,44)(45,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,22,23,19)(18,24,26,27)(20,29,21,28)(25,31,30,32)(33,36)(34,40)(35,44)(37,46)(38,45)(39,48)(41,43)(42,47);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,18)(19,24)(20,25)(21,30)(22,27)(23,26)(28,31)(29,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571615);