#453827

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

568 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)(11,21)(12,26,30,36)(13,19)(14,18)(15,20)(16,24)(17,29,35,28)(22,31,27,34)(23,32)(25,33)(37,39,40,38);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,33)(19,21)(20,32)(22,30)(26,37)(28,39)(29,38)(31,35)(36,40);(1,3,5,4)(2,7,6,8)(9,12,13,28,15,30,14,29)(10,17,18,26,20,35,19,36)(11,22,24,39,25,27,23,38)(16,31,21,37,32,34,33,40);(1,4,5,3)(2,7,6,8)(10,18,20,19)(11,23,25,24)(16,32)(17,26,35,36)(21,33)(22,38,27,39)(31,34)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361023);