#423148

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=ef,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=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, 330344);