#423215

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	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1604 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)(9,10,15,20)(11,21,25,32)(12,26,30,35)(13,19,14,18)(16,24,33,23)(17,28,36,29)(22,38,40,31)(27,34,39,37);(5,6)(7,8)(9,11,13,23,15,25,14,24)(10,16,18,21,20,33,19,32)(12,27,28,40,30,39,29,22)(17,34,35,38,36,37,26,31);(1,2,3,4)(6,8)(9,12)(10,17)(11,22)(13,29)(14,28)(15,30)(16,31)(18,26)(19,35)(20,36)(21,37)(23,39)(24,27)(25,40)(32,34)(33,38);(11,23)(13,14)(16,21)(18,19)(22,39)(24,25)(26,35)(27,40)(28,29)(31,37)(32,33)(34,38)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=ef,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=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

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, 330411);