#8417477

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
prime radical	#42608	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324673);