#150815

order	512
nilpotency class	5
isoclinism	isoclinic class of #150812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

282 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gh,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efg,g^b=gi,d^c=dhi,e^c=ei,f^c=fi,e^d=eh,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58011);