#241359

order	512
nilpotency class	3
isoclinism	isoclinic class of #233984
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

800 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148555);