#6993534

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1393 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=fh,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900730);