#355933

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303246, #72303296, #72303318, …
maximal normal subgroup of	#72303246, #72303296, #72303318, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263129);