#448330

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1199 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=egi,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=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, 355526);