#451271

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

825 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358467);