#6529777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(21,22,27,32)(23,28,34,42)(24,37,39,46)(25,31,26,30)(29,40,45,38)(33,47,48,52)(35,44,36,43)(41,50,51,49);(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,28)(24,33)(25,35)(26,36)(27,34)(29,41)(30,43)(31,44)(32,42)(37,47)(38,49)(39,48)(40,50)(45,51)(46,52);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,29)(23,33)(25,38)(26,40)(27,39)(28,41)(30,37)(31,46)(32,45)(34,48)(35,49)(36,50)(42,51)(43,47)(44,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,34)(28,42)(33,48)(35,36)(41,51)(43,44)(47,52)(49,50);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,35,34,36)(24,38,39,40)(28,43,42,44)(29,37,45,46)(33,49,48,50)(41,47,51,52)

Polycyclic presentation:

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

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, 6436973);