#493261

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	C₂³	C₂²×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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)(4,8)(5,6)(9,10,15,21,17,22,33,13,16,23,32,19,14,20,12,31)(11,24,28,34,30,18,25,36,29,37,35,40,27,38,39,26);(2,6)(7,8)(9,11,16,29)(10,18)(12,25,33,39)(13,34)(14,27,17,30)(15,35,32,28)(19,36)(20,24)(21,26)(22,37)(23,38)(31,40);(1,3,5,4)(2,7,6,8)(9,12,17,15,16,33,14,32)(10,19,20,13,23,21,22,31)(11,25,30,35,29,39,27,28)(18,36,24,34,38,26,37,40);(1,4,5,3)(2,8,6,7)(9,13,12,22,14,21,32,10,16,31,33,20,17,19,15,23)(11,26,39,38,27,40,35,37,29,36,25,18,30,34,28,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eg,d^2=ef,e^2=i,f^2=h,h^2=i,b^a=be,c^a=ch,g^a=gi,d^b=dh,f^b=fi,d^c=dh,f^c=fi,g^d=gi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313240, #72313292, #72313298, …
maximal normal subgroup of	#72313240, #72313292, #72313298, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400457);