#6404401

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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:

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

Polycyclic presentation:

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

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, 6311597);