#402075

order	512
nilpotency class	3
isoclinism	isoclinic class of #396113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 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:

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

Polycyclic presentation:

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

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, 309271);