#417786

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2766 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)(11,23)(12,27)(13,19)(14,21)(15,20)(16,22)(17,25)(18,29)(24,38)(26,33)(28,32)(30,36)(31,35)(34,40)(37,39);(1,2,3,4)(5,6)(7,8)(9,11,14,25,16,13,15,26)(10,17,20,23,22,33,21,19)(12,24,29,39,30,28,31,40)(18,32,27,37,35,38,36,34);(6,8)(9,12)(10,18)(11,24)(13,28)(14,29)(15,31)(16,30)(17,32)(19,34)(20,27)(21,36)(22,35)(23,37)(25,39)(26,40)(33,38);(2,4)(9,13,16,11)(10,19,22,23)(12,28,30,24)(14,25,15,26)(17,20,33,21)(18,34,35,37)(27,38,36,32)(29,39,31,40)

Polycyclic presentation:

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

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, 324982);