#441143

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1460 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,d^2=e,e^2=i,f^2=i,g^2=i,h^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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi

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, 348339);