#245699

order	512
nilpotency class	3
isoclinism	isoclinic class of #245615
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#25773	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1213 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152895);