#8096337

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(5,8)(6,9)(7,14)(11,13)(15,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,40,43)(35,44,41,36)(37,47,48,39)(38,42,46,45);(1,3,6,8)(2,5,9,10)(4,11,14,15)(7,13,12,16)(20,28)(25,30)(27,29)(31,32)(33,35,40,41)(34,36,43,44)(37,46,48,38)(39,45,47,42);(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,36,40,44)(34,41,43,35)(37,42,48,45)(38,39,46,47);(1,4)(2,7)(3,11)(5,13)(6,14)(8,15)(9,12)(10,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,37)(34,39)(35,38)(36,42)(40,48)(41,46)(43,47)(44,45);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,38)(34,42)(35,37)(36,39)(40,46)(41,48)(43,45)(44,47);(1,5)(2,8)(3,9)(4,13)(6,10)(7,15)(11,12)(14,16)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,39)(34,37)(35,45)(36,46)(38,44)(40,47)(41,42)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003533);