#6535075

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3565	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2737 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)(21,22,27,32)(23,33,35,44)(24,38,41,45)(25,30,26,31)(28,36,43,37)(29,40,46,39)(34,48,49,42)(47,50,52,51);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,35)(22,28,32,43)(24,34,41,49)(25,36,26,37)(29,42,46,48)(30,33,31,44)(38,52,45,47)(39,50,40,51);(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,34)(25,39)(26,40)(27,41)(28,42)(30,45)(31,38)(32,46)(33,47)(35,49)(36,50)(37,51)(43,48)(44,52);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(23,35)(28,43)(33,44)(34,49)(36,37)(42,48)(47,52)(50,51);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,39,41,40)(28,33,43,44)(29,45,46,38)(34,50,49,51)(42,47,48,52)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6442271);