#8917899

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	#55107	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8825095);