#463534

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370730);