#8119905

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	#37912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,36)(37,40)(38,45)(39,41)(42,47)(43,48)(44,46);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35,40,39)(34,41,37,36)(38,43,47,46)(42,48,45,44);(2,5)(8,10)(9,12)(15,16)(20,28)(23,30)(26,32)(29,31)(33,36)(34,39)(35,37)(38,44)(40,41)(42,46)(43,45)(47,48);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,37,40,34)(35,41,39,36)(38,45,47,42)(43,48,46,44);(1,5,7,2)(3,10,6,8)(4,12,14,9)(11,16,13,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48);(1,6,7,3)(2,10,5,8)(4,13,14,11)(9,16,12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,39,40,35)(34,41,37,36)(38,46,47,43)(42,48,45,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8027101);