#8340100

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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)(3,5)(4,12)(6,7)(8,10)(9,15)(11,16)(13,14)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(2,7)(3,10)(9,12)(11,13)(19,20)(24,26)(27,28)(31,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,7)(5,8)(9,12)(14,16)(17,19,23,20)(18,24,21,26)(22,28,30,27)(25,32,29,31)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=i,e^2=h,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, 8247296);