#423632

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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	#23426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1634 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330828);