#8417005

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	#52873	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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)(3,11)(4,9)(5,10)(6,8)(7,14)(12,16)(13,15)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,13,16,9)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(34,35)(40,43)(41,42)(47,48);(2,8)(5,16)(7,11)(9,13)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,11,16,7)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,3)(2,7)(4,10)(5,13)(6,14)(8,11)(9,16)(12,15)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,11,15,7)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gi,e^2=i,f^2=g,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, 8324201);