#6373975

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60673	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 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)(9,10,14,18,15,19,13,29)(11,20,23,31,24,36,22,16)(12,17,28,34,25,32,27,40)(21,35,38,33,26,39,37,30);(2,5)(6,8)(12,25)(17,32)(21,26)(27,28)(30,33)(34,40)(35,39)(37,38);(9,11,13,22,15,24,14,23)(10,16,18,20,19,31,29,36)(12,26,27,38,25,21,28,37)(17,33,34,39,32,30,40,35);(9,12,14,28,15,25,13,27)(10,17,18,34,19,32,29,40)(11,21,23,38,24,26,22,37)(16,30,20,35,31,33,36,39);(1,3,4,7)(2,6,5,8)(12,25)(17,32)(21,26)(27,28)(30,33)(34,40)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=di,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281171);