#6386258

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2792 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,31)(24,35)(25,29)(26,39)(27,33)(28,37)(30,38)(32,47)(34,42)(36,44)(40,43)(41,51)(45,49)(46,50)(48,52);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,25,33)(22,27,29,31)(24,32,36,48)(26,34,38,50)(28,41,43,45)(30,42,39,46)(35,52,44,47)(37,49,40,51);(21,24,25,36)(22,28,29,43)(23,32,33,48)(26,40,38,37)(27,41,31,45)(30,44,39,35)(34,51,50,49)(42,47,46,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,33)(27,31)(32,48)(34,50)(41,45)(42,46)(47,52)(49,51);(6,12)(10,13)(11,20)(18,19)(22,29)(23,33)(28,43)(30,39)(32,48)(34,50)(35,44)(49,51)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293454);