#346993

order	512
nilpotency class	3
isoclinism	isoclinic class of #346950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62847	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2169 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)(3,4)(7,8)(9,10,14,19)(11,15,22,32)(12,17,23,34)(13,25,28,18)(16,27,33,24)(20,36,37,30)(21,35,26,31)(29,39,40,38);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,30)(18,35)(19,32)(21,28)(23,37)(24,39)(25,31)(27,38)(33,40)(34,36);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,20)(13,27)(14,23)(15,29)(17,25)(18,34)(19,33)(21,39)(22,37)(24,28)(26,38)(30,31)(32,40)(35,36);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,20)(13,24)(14,23)(15,30)(16,25)(18,33)(19,34)(21,38)(22,37)(26,39)(27,28)(29,31)(32,36)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254189);