#410948

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
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

759 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)(18,30)(23,32)(26,33)(28,40)(29,37)(31,36)(34,38)(35,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16,20,33)(12,23)(13,24)(14,26)(15,25)(17,19,22,21)(18,32,37,40)(27,34,36,35)(28,31)(29,38)(30,39);(9,11)(10,17,20,22)(12,28)(13,24)(14,26)(15,25)(16,21,33,19)(18,35,37,34)(23,31)(27,32,36,40)(29,39)(30,38);(1,4,5,3)(2,6,7,8)(9,12,13,29,15,31,14,30)(10,18,19,36,20,37,21,27)(11,23,24,38,25,28,26,39)(16,32,22,35,33,40,17,34)

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,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306970, #72307320, #72307328
maximal normal subgroup of	#72306970, #72307320, #72307328

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318144);