#7623028

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623027
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3584	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,8,14)(3,15,19,13)(4,20)(5,11,25,39)(6,28,16,12)(7,9,30,33)(10,34)(17,38)(18,44,48,41)(21,49)(22,43)(23,42)(24,31)(26,54,45,40)(27,32,53,57)(29,36)(35,58)(37,47)(46,59)(50,56)(51,63)(52,62)(55,60)(61,64);(1,3,16,30,8,19,6,7)(2,9,28,13,14,33,12,15)(4,17,23,47,24,34,22,29)(5,18,45,53,25,48,26,27)(10,31,37,42,38,20,36,43)(11,32,54,41,39,57,40,44)(21,46,52,64,50,58,51,55)(35,56,61,62,59,49,60,63);(1,4)(2,10)(3,17)(5,21)(6,22)(7,29)(8,24)(9,31)(11,35)(12,36)(13,42)(14,38)(15,43)(16,23)(18,46)(19,34)(20,33)(25,50)(26,51)(27,55)(28,37)(30,47)(32,56)(39,59)(40,60)(41,62)(44,63)(45,52)(48,58)(49,57)(53,64)(54,61);(1,5)(2,11)(3,18)(4,21)(6,26)(7,27)(8,25)(9,32)(10,35)(12,40)(13,41)(14,39)(15,44)(16,45)(17,46)(19,48)(20,49)(22,51)(23,52)(24,50)(28,54)(29,55)(30,53)(31,56)(33,57)(34,58)(36,60)(37,61)(38,59)(42,62)(43,63)(47,64);(5,25)(11,39)(18,48)(21,50)(26,45)(27,53)(32,57)(35,59)(40,54)(41,44)(46,58)(49,56)(51,52)(55,64)(60,61)(62,63)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,f^2=i,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gh,h^a=hi,g^c=ghi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530224);