#423859

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1328 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331055);