#234874

order	512
nilpotency class	3
isoclinism	isoclinic class of #234865
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60949	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1312 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)(11,15)(12,16)(13,17)(14,18)(19,26)(20,27)(21,28)(22,30)(23,31)(24,32)(25,33)(29,35)(34,38)(36,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,21)(10,15,18,28)(12,19,23,35)(13,24,25,20)(16,29,31,26)(17,32,33,27)(22,36,37,34)(30,38,40,39);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,30)(18,31)(20,34)(21,35)(24,36)(25,37)(27,38)(28,29)(32,39)(33,40);(9,13,14,25)(10,17,18,33)(11,20,21,24)(12,22,23,37)(15,27,28,32)(16,30,31,40)(19,34,35,36)(26,38,29,39)

Polycyclic presentation:

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

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, 142070);