#8662894

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	#54868	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8570090);