#409651

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

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	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

856 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)(11,15,21,28)(12,16)(13,17)(14,18)(19,25)(20,30,34,26)(22,27,36,29)(23,32)(24,33)(31,37)(35,38,40,39);(2,5)(6,8)(9,11,13,22,14,21,25,36)(10,15,19,29,18,28,17,27)(12,20,24,35,23,34,37,40)(16,30,31,39,32,26,33,38);(9,12)(10,16)(11,20)(13,24)(14,23)(15,26)(17,33)(18,32)(19,31)(21,34)(22,35)(25,37)(27,39)(28,30)(29,38)(36,40);(1,3,4,7)(2,6,5,8)(10,17,18,19)(12,23)(15,27,28,29)(16,31,32,33)(20,34)(24,37)(26,38,30,39)(35,40)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 316847);