#128583

order	512
nilpotency class	4
isoclinism	isoclinic class of #128583
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8174
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,14,17,39)(4,9)(5,23)(6,11)(7,13)(8,16,28,44)(10,24)(12,26)(15,38,43,60)(18,40,46,41)(19,47,49,61)(20,50)(21,34)(22,37)(25,35)(27,42,55,62)(29,45,30,57)(31,48,56,63)(32,52)(33,53)(36,54)(51,59)(58,64);(1,3)(2,8,11,29,13,28,12,30)(4,19,21,49)(5,16)(6,18)(7,17)(9,31,33,27,34,56,37,55)(10,14,35,40,36,39,23,41)(15,22,43,53)(20,48,51,63)(24,45)(25,44)(26,46)(32,47,50,38,58,61,59,60)(42,52,62,64)(54,57);(1,4)(2,9)(3,15,17,43)(5,20)(6,22)(7,21)(8,27,28,55)(10,32)(11,37)(12,33)(13,34)(14,38,39,60)(16,42,44,62)(18,19,46,49)(23,50)(24,52)(25,51)(26,53)(29,31,30,56)(35,59)(36,58)(40,47,41,61)(45,48,57,63)(54,64)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,f^2=h,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=cgh,e^b=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35779);