#6388821

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22)(23,28)(24,35)(25,30)(26,39)(27,31)(29,36)(32,47)(33,42)(34,43)(37,45)(38,44)(40,46)(41,50)(48,52)(49,51);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23)(22,28)(24,32)(25,38)(26,34)(27,33)(29,41)(30,44)(31,42)(35,47)(36,50)(37,48)(39,43)(40,49)(45,52)(46,51);(21,24,27,37)(22,29,31,46)(23,32,33,48)(25,36,26,40)(28,41,42,51)(30,45,39,35)(34,49,38,50)(43,47,44,52);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(23,33)(28,42)(32,48)(34,38)(41,51)(43,44)(47,52)(49,50);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,25,27,26)(22,30,31,39)(23,34,33,38)(24,36,37,40)(28,43,42,44)(29,45,46,35)(32,49,48,50)(41,47,51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296017);