#493319

order	512
nilpotency class	2
isoclinism	isoclinic class of #180873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313488, #72313527, #72313529, …
maximal normal subgroup of	#72313488, #72313527, #72313529, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400515);