#408426

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

731 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315622);