#138379

order	512
nilpotency class	4
isoclinism	isoclinic class of #125508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gi,d^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45575);