#291204

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=fhi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198400);