#6465963

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#28247	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=hi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373159);