#456627

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1391 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:

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

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363823);