#6628674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=gh,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi

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, 6535870);