#8096165

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	#45693	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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)(3,10)(4,12)(5,7)(6,9)(8,14)(11,15)(13,16)(17,18)(19,21)(20,25)(22,23)(24,26)(27,30)(28,32)(29,31)(33,34)(35,38)(36,40)(37,41)(39,44)(42,46)(43,47)(45,48);(1,3,6,5)(2,7,9,10)(4,11,14,13)(8,15,12,16)(18,23)(19,26)(25,31)(27,32)(33,35,37,43)(34,38,41,47)(36,42,44,48)(39,45,40,46);(17,19)(18,24)(20,27)(21,23)(22,26)(25,28)(29,32)(30,31)(33,36)(34,39)(35,42)(37,44)(38,45)(40,41)(43,48)(46,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,28)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,46)(41,47)(44,48);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,31)(24,28)(26,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,41)(43,48)(45,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,21,22,24)(18,19,23,26)(20,28,29,30)(25,32,31,27)(33,34)(35,38)(36,39)(37,41)(40,44)(42,45)(43,47)(46,48)

Polycyclic presentation:

2^9|b^2=gh,e^2=gi,f^2=i,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, 8003361);