#263379

order	512
nilpotency class	3
isoclinism	isoclinic class of #263379
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18190	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

901 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,d^c=dh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170575);