#465860

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1088 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,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, 373056);