#8733942

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

4235 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8641138);