#6971666

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,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, 6878862);