#8718229

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	#56332	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625425);