#135759

order	512
nilpotency class	4
isoclinism	isoclinic class of #135758
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1844
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#492
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224817, #72224824, #72224825, …
maximal normal subgroup of	#72224817, #72224824, #72224825, …
maximal quotient of	#72225164, #72225166

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42955);