#6563319

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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

1459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6470515);