#415164

order	512
nilpotency class	3
isoclinism	isoclinic class of #395944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

830 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)(4,5)(6,8)(9,10,15,19,14,18,13,20)(11,21,24,31,25,35,23,16)(12,17,27,36,28,33,29,34)(22,37,26,30,38,32,39,40);(9,11,13,23,14,25,15,24)(10,16,19,21,18,31,20,35)(12,26,27,38,28,39,29,22)(17,32,34,30,33,37,36,40);(2,5)(6,7)(9,12,13,27,14,28,15,29)(10,17,19,34,18,33,20,36)(11,22,23,26,25,38,24,39)(16,30,21,37,31,40,35,32);(1,3,4,8)(2,6,5,7)(10,18)(16,31)(17,33)(19,20)(21,35)(30,40)(32,37)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322360);