#425701

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

875 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)(3,6)(4,8)(5,7)(9,10,14,20,15,21,16,22)(11,23,26,35,27,36,28,17)(12,19,30,40,13,18,32,38)(24,39,29,33,25,37,31,34);(1,3,5,4)(2,6,7,8)(9,11,15,27)(10,17,21,35)(12,29,13,31)(14,28,16,26)(18,37,19,39)(20,36,22,23)(24,32,25,30)(33,40,34,38);(1,4,5,3)(2,6,7,8)(9,12,15,13)(10,18,21,19)(11,24,27,25)(14,30,16,32)(17,33,35,34)(20,38,22,40)(23,37,36,39)(26,29,28,31);(9,13,15,12)(10,19,21,18)(11,25,27,24)(14,30,16,32)(17,34,35,33)(20,38,22,40)(23,37,36,39)(26,29,28,31)

Polycyclic presentation:

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

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, 332897);