#492861

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1552 conjugacy classes of subgroups; only the 866 normal subgroups are shown.

normal subgroups only

This group has 1552 conjugacy classes of subgroups — too many to draw. Only its 866 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,10,16,12,19,30,18)(7,20,23,46,25,47,52,35)(8,14,28,40,26,38,56,42)(9,17,11,15,31,43,32,44)(13,24,34,53,37,54,36,22)(21,45,50,65,48,64,67,60)(27,41,29,39,55,63,57,62)(33,51,59,66,58,68,61,49);(2,4)(5,7,11,24,10,23,31,53,12,25,32,54,30,52,9,22)(6,13,17,35,18,36,44,47,19,37,43,46,16,34,15,20)(8,21,29,51,28,50,55,66,26,48,57,68,56,67,27,49)(14,33,41,60,42,61,62,64,38,58,63,65,40,59,39,45);(5,8)(6,14)(7,21)(9,27)(10,28)(11,29)(12,26)(13,33)(15,39)(16,40)(17,41)(18,42)(19,38)(20,45)(22,49)(23,50)(24,51)(25,48)(30,56)(31,55)(32,57)(34,59)(35,60)(36,61)(37,58)(43,63)(44,62)(46,65)(47,64)(52,67)(53,66)(54,68);(8,26)(14,38)(21,48)(27,55)(28,56)(29,57)(33,58)(39,62)(40,42)(41,63)(45,64)(49,66)(50,67)(51,68)(59,61)(60,65)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=ghi,h^a=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400057);