#135296

order	512
nilpotency class	4
isoclinism	isoclinic class of #124940
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

534 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,21,23,16)(12,26,28,17)(13,20)(14,30)(15,18)(19,29,34,27)(22,31)(24,37,25,33)(32,39)(35,40)(36,38);(2,5)(6,8)(9,11)(10,16)(12,22)(13,24)(14,25)(15,23)(17,31)(18,21)(19,32)(20,33)(26,35)(27,38)(28,40)(29,39)(30,37)(34,36);(1,3,4,7)(2,6,5,8)(10,17,18,26)(12,27)(13,14)(16,31,21,35)(19,20,34,30)(22,38)(24,25)(28,29)(32,33,36,37)(39,40)

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,f^b=fi,e^c=eh,f^c=fgh,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307712
maximal normal subgroup of	#72307712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42492);