#217875

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

668 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)(9,10,15,19)(11,16,12,21)(13,18,14,23)(17,20,24,22)(25,26,29,34,31,28,30,33)(27,32,35,40,37,38,36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,15)(13,22)(14,20)(17,18)(19,21)(23,24)(25,27,30,36,31,37,29,35)(26,32,33,39,28,38,34,40);(3,8)(5,7)(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(25,28,31,26)(27,32,37,38)(29,33,30,34)(35,40,36,39);(2,6)(5,7)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(25,29,31,30)(26,33,28,34)(27,35,37,36)(32,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,d^b=dhi,d^c=di,f^c=fh

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, 125071);