#130044

order	512
nilpotency class	4
isoclinism	isoclinic class of #130043
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8344
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

568 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37240);