#150033

order	512
nilpotency class	5
isoclinism	isoclinic class of #149894
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

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

normal subgroups only

This group has 229 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:

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

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,c^a=ci,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

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57229);