#245802

order	512
nilpotency class	3
isoclinism	isoclinic class of #245711
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	C₂×C₄²	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1132 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:

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

Polycyclic presentation:

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

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, 152998);