#6672361

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1705 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,26,30)(23,27,33,41)(24,28,35,43)(25,29,37,45)(31,39,38,46)(32,40,48,51)(34,42,36,44)(47,50,49,52);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(23,31)(24,34)(25,37)(27,39)(28,42)(29,45)(32,48)(33,38)(35,36)(40,51)(41,46)(43,44);(2,4)(21,23,26,33)(22,27,30,41)(24,32,35,48)(25,38,37,31)(28,40,43,51)(29,46,45,39)(34,49,36,47)(42,52,44,50);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,35)(28,43)(32,48)(34,36)(40,51)(42,44)(47,49)(50,52);(7,15)(10,13)(14,20)(18,19)(21,24,26,35)(22,28,30,43)(23,32,33,48)(25,34,37,36)(27,40,41,51)(29,42,45,44)(31,47,38,49)(39,50,46,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579557);