#6436149

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1752 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343345);