#230981

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

594 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=eh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ch,d^b=dei,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, 138177);