#410947

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

773 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)(3,8)(4,6)(5,7)(9,10)(11,22)(12,27)(13,21)(14,19)(15,20)(16,24)(17,25)(18,29)(23,31)(26,32)(28,35)(30,34)(33,37)(36,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16,20,32)(12,23)(13,24)(14,26)(15,25)(17,19,22,21)(18,31,35,40)(27,33,34,36)(28,37)(29,39)(30,38);(9,11)(10,17,20,22)(12,23)(13,24)(14,26)(15,25)(16,21,32,19)(18,33,35,36)(27,40,34,31)(28,37)(29,39)(30,38);(1,4,5,3)(2,6,7,8)(9,12,13,28,15,30,14,29)(10,18,19,34,20,35,21,27)(11,23,24,37,25,38,26,39)(16,31,22,36,32,40,17,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318143);