#150032

order	512
nilpotency class	5
isoclinism	isoclinic class of #149893
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2122
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	#11576
prime radical	#525	C₂×C₄
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

232 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=g,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=cg,d^b=dh,e^b=efi,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57228);