#411521

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

678 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:

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

Polycyclic presentation:

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

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, 318717);