#468002

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

648 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)(11,17)(12,23)(13,27,30,18)(14,32)(15,20)(16,22)(19,24)(21,35,25,39)(26,34)(28,33,38,40)(29,36,31,37);(9,11,14,22,15,12,24,26)(10,16,19,17,20,34,32,23)(13,21,31,28,30,25,29,38)(18,33,37,35,27,40,36,39);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,15)(13,28)(14,22)(16,19)(18,33)(20,23)(21,31)(24,26)(25,29)(27,40)(30,38)(32,34)(35,36)(37,39);(1,4,5,3)(2,6,7,8)(9,13,15,30)(10,18,20,27)(11,21,12,25)(14,31,24,29)(16,33,34,40)(17,35,23,39)(19,37,32,36)(22,28,26,38)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 375198);