#6450380

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle, prime radical	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(18,23)(24,25)(26,32)(27,29);(2,5)(6,8)(17,18,21,23)(19,27,20,29)(22,32,31,26)(24,30,25,28);(10,13)(14,16)(17,19,21,20)(18,24,23,25)(22,30,31,28)(26,29,32,27);(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,25,23,24)(22,28,31,30)(26,27,32,29);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357576);