#8340042

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	#55410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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)(3,5)(4,12)(6,7)(8,10)(9,15)(11,16)(13,14)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(2,7)(3,10)(9,12)(11,13)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(2,7)(5,8)(9,12)(14,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,37,38,35)(34,41,42,39)(36,44,45,43)(40,47,48,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=ghi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8247238);