#217697

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
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

675 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)(9,10,14,18)(11,15,20,24)(12,17,13,22)(16,21,23,19)(25,26,29,33,30,28,35,32)(27,34,38,40,37,31,36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24)(25,27,35,36,30,37,29,38)(26,31,32,40,28,34,33,39);(3,8)(5,7)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,28)(26,30)(27,31)(29,32)(33,35)(34,37)(36,40)(38,39);(2,6)(5,7)(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)(25,29,30,35)(26,32,28,33)(27,36,37,38)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,d^c=di,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124893);