#333002

order	512
nilpotency class	3
isoclinism	isoclinic class of #332983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

726 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=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, 240198);