#8926596

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

3784 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)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(34,36)(35,43)(38,48)(41,46);(1,3)(2,8)(4,12)(5,9)(6,11)(7,13)(10,15)(14,16)(21,29)(24,28)(26,32)(30,31)(33,34,39,36)(35,44,43,40)(37,46,45,41)(38,42,48,47);(1,4,7,15)(2,9,11,16)(3,12,13,10)(5,6,14,8)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35,39,43)(34,40,36,44)(37,38,45,48)(41,42,46,47);(1,5)(2,10)(3,9)(4,6)(7,14)(8,15)(11,12)(13,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36,39,34)(35,44,43,40)(37,46,45,41)(38,47,48,42);(1,6)(2,3)(4,14)(5,15)(7,8)(9,12)(10,16)(11,13)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37,39,45)(34,41,36,46)(35,38,43,48)(40,42,44,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,11)(12,15)(14,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,38)(34,42)(35,45)(36,47)(37,43)(39,48)(40,46)(41,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833792);