#456194

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

1088 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363390);