#214515

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#35130	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

963 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)(9,10,14,16,13,17,12,21)(11,18,19,23,20,15,22,24)(25,26)(27,34,36,39)(28,37)(29,32)(30,33)(31,35,40,38);(1,3)(2,5)(4,8)(6,7)(9,11,13,20)(10,15,17,18)(12,22,14,19)(16,24,21,23);(3,8)(5,7)(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,40)(34,37);(2,6)(5,7)(9,12,13,14)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,f^2=i,g^2=h,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, 121711);