#316995

order	512
nilpotency class	3
isoclinism	isoclinic class of #316991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28225	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

613 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,g^c=gi

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, 224191);