#493301

order	512
nilpotency class	2
isoclinism	isoclinic class of #180869
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	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

569 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)(3,6)(4,5)(7,8)(9,10,14,21,16,22,12,19,17,23,15,32,13,20,30,39)(11,24,27,34,28,18,33,35,29,37,31,40,26,38,25,36);(2,5)(3,7)(9,11,15,27,16,28,30,33,17,29,14,31,13,26,12,25)(10,18,21,36,22,37,19,34,23,38,32,35,20,24,39,40);(1,3,4,7)(2,6,5,8)(9,12,13,14,17,30,16,15)(10,19,20,21,23,39,22,32)(11,25,26,31,29,33,28,27)(18,34,24,36,38,40,37,35);(2,5)(6,8)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313260, #72313282, #72313298, …
maximal normal subgroup of	#72313260, #72313282, #72313298, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400497);