#317188

order	512
nilpotency class	3
isoclinism	isoclinic class of #317171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 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)(3,7)(4,6)(5,8)(9,10,15,18)(11,19,22,37)(12,17,27,35)(13,14,28,29)(16,30,32,21)(20,33,39,36)(23,31,38,40)(24,25,34,26);(9,11,14,21,15,22,29,30)(10,16,13,19,18,32,28,37)(12,23,26,39,27,38,25,20)(17,33,34,40,35,36,24,31);(2,6)(3,4)(7,8)(9,12)(10,17)(11,20)(13,24)(14,25)(15,27)(16,31)(18,35)(19,36)(21,38)(22,39)(23,30)(26,29)(28,34)(32,40)(33,37);(1,3,5,4)(2,7,8,6)(9,13,14,18,15,28,29,10)(11,19,21,32,22,37,30,16)(12,24,25,35,27,34,26,17)(20,36,38,40,39,33,23,31)

Polycyclic presentation:

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

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, 224384);