#453047

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

693 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:

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

Polycyclic presentation:

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

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, 360243);