#8522133

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8429329);