#8718314

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4023 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18)(19,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32)(33,34,39,36)(35,40,41,37)(38,42,47,44)(43,46,48,45);(2,7)(3,12)(9,14)(10,13)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35,39,41)(34,40,36,37)(38,43,47,48)(42,46,44,45);(2,7)(3,12)(9,14)(10,13)(20,29)(23,28)(26,32)(30,31)(34,36)(35,41)(42,44)(43,48);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,16)(10,15)(17,20)(18,23)(19,26)(21,29)(22,30)(24,31)(25,28)(27,32)(33,36,39,34)(35,40,41,37)(38,42,47,44)(43,45,48,46);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32)(33,37,39,40)(34,41,36,35)(38,46,47,45)(42,43,44,48);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,18)(19,22)(20,28)(21,25)(23,29)(24,27)(26,31)(30,32)(33,38,39,47)(34,42,36,44)(35,43,41,48)(37,45,40,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=h,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, 8625510);