#452184

order	512
nilpotency class	3
isoclinism	isoclinic class of #417542
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1278 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)(3,6,4,8)(9,10)(11,19,22,15)(12,16)(13,25,26,17)(14,18)(20,29,34,27)(21,32)(23,36,38,30)(24,31)(28,35)(33,40)(37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,32)(18,19)(23,37)(24,34)(25,28)(26,35)(27,31)(30,40)(33,38)(36,39);(2,7)(6,8)(9,12)(10,16)(11,20)(13,23)(14,24)(15,27)(17,30)(18,31)(19,29)(21,33)(22,34)(25,36)(26,38)(28,39)(32,40)(35,37);(1,4,5,3)(2,8,7,6)(9,13,14,26)(10,17,18,25)(11,21,22,35)(12,23,24,38)(15,28,19,32)(16,30,31,36)(20,33,34,37)(27,39,29,40)

Polycyclic presentation:

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

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, 359380);