#423672

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1598 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330868);