#294854

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 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)(4,5)(6,8)(9,10,14,17)(11,15,13,22)(12,19,20,24)(16,18,23,21)(25,26,29,32)(27,34,35,31)(28,36,37,40)(30,39,38,33);(2,5)(6,7)(9,11,14,13)(10,15,17,22)(12,18,20,21)(16,19,23,24)(25,27,29,35)(26,31,32,34)(28,30,37,38)(33,40,39,36);(9,12,13,21,14,20,11,18)(10,16,15,19,17,23,22,24)(25,28,30,35,29,37,38,27)(26,31,33,36,32,34,39,40);(1,3,4,8)(2,6,5,7)(12,20)(16,23)(18,21)(19,24)(26,32)(31,34)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202050);