#419187

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1892 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=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, 326383);