#217981

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

596 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,17)(13,22)(14,18)(16,19)(20,24)(21,23)(25,26,31,33,30,28,29,38)(27,34,35,39,36,32,37,40);(2,5)(3,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24)(25,27,31,35,30,36,29,37)(26,32,33,40,28,34,38,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,28,30,26)(27,32,36,34)(29,33,31,38)(35,40,37,39);(2,5)(6,8)(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)(25,29,30,31)(26,33,28,38)(27,35,36,37)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,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, 125177);