#6481733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,28,33)(23,34,24,38)(25,30,40,45)(26,32,27,31)(29,37,44,36)(35,48,39,51)(41,47,42,46)(43,50,52,49);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,28,24)(22,29,33,44)(25,35,40,39)(26,37,27,36)(30,43,45,52)(31,38,32,34)(41,50,42,49)(46,51,47,48);(7,13)(10,17)(14,15)(18,19)(21,24,28,23)(22,29,33,44)(25,39,40,35)(26,36,27,37)(30,43,45,52)(31,38,32,34)(41,49,42,50)(46,51,47,48);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,25,28,40)(22,30,33,45)(23,35,24,39)(26,41,27,42)(29,43,44,52)(31,46,32,47)(34,48,38,51)(36,49,37,50);(6,12)(10,17)(11,20)(18,19)(25,40)(30,45)(35,39)(41,42)(43,52)(46,47)(48,51)(49,50)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388929);