#453803

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

727 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)(6,8)(9,10,15,19,14,18,28,34)(11,12,16,23,21,24,29,35)(13,17,27,33,26,32,30,36)(20,37,38,40,25,22,31,39);(1,2,4,5)(3,6,7,8)(9,11,14,21)(10,12,18,24)(13,25,26,20)(15,29,28,16)(17,22,32,37)(19,35,34,23)(27,38,30,31)(33,40,36,39);(1,3,4,7)(2,6,5,8)(9,12)(10,16,18,29)(11,19,21,34)(13,22)(14,24)(15,23)(17,31,32,38)(20,36,25,33)(26,37)(27,39)(28,35)(30,40);(2,5)(6,8)(9,13,14,26)(10,17,18,32)(11,20,21,25)(12,22,24,37)(15,30,28,27)(16,31,29,38)(19,36,34,33)(23,40,35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=efh,c^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 360999);