#423217

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1510 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)(9,10,16,13)(11,21,26,23)(12,27,32,36)(14,20,15,19)(17,25,33,24)(18,30,29,31)(22,37,40,34)(28,35,39,38);(5,6)(7,8)(9,11,15,25,16,26,14,24)(10,17,20,23,13,33,19,21)(12,28,31,40,32,39,30,22)(18,35,27,37,29,38,36,34);(1,2,3,4)(6,8)(9,12)(10,18)(11,22)(13,29)(14,31)(15,30)(16,32)(17,34)(19,27)(20,36)(21,35)(23,38)(24,28)(25,39)(26,40)(33,37);(9,13,16,10)(11,23,26,21)(12,29,32,18)(14,19,15,20)(17,24,33,25)(22,38,40,35)(27,30,36,31)(28,37,39,34)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=ef,d^2=i,e^2=i,f^2=i,g^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=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 330413);