#8688322

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	#2819	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595518);