#152954

order	512
nilpotency class	5
isoclinism	isoclinic class of #152953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13224
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

297 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,14,19,38)(4,21)(5,25)(6,12,28,39)(7,13)(8,17,32,42)(9,22)(10,26,29,44)(11,15)(16,30,47,57)(18,34,49,33)(20,31,27,41)(23,35,55,56)(24,52)(36,61,51,53)(37,43)(40,62,60,46)(45,63,64,50)(48,59,54,58);(1,3,15,42,7,20,5,18)(2,8,29,41,13,34,10,38)(4,16,43,64,24,48,22,46)(6,27,26,17,28,19,44,49)(9,30,56,63,37,59,35,60)(11,31,12,33,25,14,39,32)(21,50,36,58,52,40,51,57)(23,54,53,45,55,47,61,62);(1,4,15,43,7,24,5,22)(2,9,25,52,13,37,11,21)(3,16,42,64,20,48,18,46)(6,23,44,61,28,55,26,53)(8,30,14,40,34,59,31,50)(10,35,12,36,29,56,39,51)(17,45,27,54,49,62,19,47)(32,57,38,60,33,58,41,63)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ei,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dgh,f^b=fghi,g^b=ghi,h^b=hi,f^d=fhi,g^f=gi

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, 60150);