#8739182

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

3947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=h,e^2=h,f^2=gh,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, 8646378);