#6916049

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10,15,18)(11,19,12,16)(13,22,14,17)(20,23,21,24)(25,26,30,32)(27,36,39,52)(28,41,44,34)(29,35,31,46)(33,38,47,40)(37,54,42,53)(43,51,45,50)(48,55,49,56);(1,2)(3,7)(4,6)(5,8)(9,11,15,12)(10,16,18,19)(13,20,14,21)(17,23,22,24)(26,32)(33,47)(34,41)(35,46)(36,52)(48,49)(50,51)(53,54);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,16,18,19)(13,20,14,21)(17,24,22,23)(25,27,29,38,30,39,31,40)(26,33,46,52,32,47,35,36)(28,42,45,55,44,37,43,56)(34,49,50,53,41,48,51,54);(3,8)(5,7)(9,13,15,14)(10,17,18,22)(11,20,12,21)(16,23,19,24)(25,28,30,44)(26,34,32,41)(27,37,39,42)(29,45,31,43)(33,48,47,49)(35,51,46,50)(36,53,52,54)(38,56,40,55);(2,6)(5,7)(9,14,15,13)(10,17,18,22)(11,20,12,21)(16,24,19,23)(25,29,30,31)(26,35,32,46)(27,38,39,40)(28,43,44,45)(33,36,47,52)(34,50,41,51)(37,55,42,56)(48,54,49,53)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fg,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6823245);