#214594

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35292	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1043 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)(9,10,13,19,14,18,12,17)(11,15,20,23,21,16,22,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,39)(35,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,18)(17,24)(19,23)(25,27,29,34)(26,30,32,39)(28,35,36,40)(31,37,33,38);(3,8)(5,7)(9,11)(10,16)(12,20)(13,22)(14,21)(15,18)(17,24)(19,23)(26,31)(28,36)(30,37)(32,33)(35,40)(38,39);(2,6)(5,7)(9,12,14,13)(10,17,18,19)(11,20,21,22)(15,23,16,24)(26,32)(27,34)(31,33)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi

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, 121790);