#346495

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62866	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2355 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253691);