#8366594

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8273790);