#8722914

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gi,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, 8630110);