#8174304

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	#53094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,9)(4,5)(6,14)(7,8)(10,15)(11,12)(13,16)(17,18,23,27)(19,24,29,32)(20,21,25,30)(22,31,28,26)(33,34,39,42)(35,41,38,40)(36,37,46,48)(43,47,45,44);(1,3,7,5)(2,4,8,9)(6,11,15,13)(10,12,14,16)(17,19,23,29)(18,24,27,32)(20,28,25,22)(21,26,30,31)(33,35,39,38)(34,40,42,41)(36,43,46,45)(37,44,48,47);(2,8)(4,9)(10,14)(12,16)(17,20,23,25)(18,21,27,30)(19,22,29,28)(24,31,32,26)(33,36)(34,37)(35,43)(38,45)(39,46)(40,44)(41,47)(42,48);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,14)(15,16)(17,20)(18,21)(19,28)(22,29)(23,25)(24,26)(27,30)(31,32)(33,37,39,48)(34,36,42,46)(35,44,38,47)(40,43,41,45);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,21)(18,25)(19,26)(20,27)(22,24)(23,30)(28,32)(29,31)(33,36)(34,37)(35,43)(38,45)(39,46)(40,44)(41,47)(42,48);(1,6,7,15)(2,10,8,14)(3,11,5,13)(4,12,9,16)(17,22,23,28)(18,26,27,31)(19,20,29,25)(21,24,30,32)(33,38,39,35)(34,41,42,40)(36,45,46,43)(37,47,48,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8081500);