#8720638

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=gi,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, 8627834);