#440768

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1177 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 347964);