#441154

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
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

1406 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)(9,10,15,20,16,21,14,22)(11,23,25,17,26,19,13,31)(12,18,28,35,29,36,30,34)(24,33,27,32,39,37,38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,17)(12,27)(13,14)(15,25)(16,26)(18,33)(19,20)(21,31)(22,23)(24,30)(28,39)(29,38)(32,35)(34,40)(36,37);(1,3,5,4)(2,7,6,8)(9,12,14,28,16,29,15,30)(10,18,20,34,21,36,22,35)(11,24,25,38,26,39,13,27)(17,32,23,37,31,40,19,33);(1,4,5,3)(2,7,6,8)(9,13,15,26,16,25,14,11)(10,19,22,31,21,23,20,17)(12,27,30,39,29,38,28,24)(18,33,35,40,36,37,34,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,d^2=ei,e^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,f^c=fi,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, 348350);