#230677

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

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

530 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)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,22,24,23)(25,26,31,27,30,33,29,35)(28,32,37,34,38,40,39,36);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,20)(15,23)(16,18)(17,22)(21,24)(25,27,31,33,30,35,29,26)(28,36,37,32,38,34,39,40);(1,3,5,4)(2,7,6,8)(10,15)(11,19)(13,21)(14,22)(16,17)(18,23)(25,28,30,38)(26,32,33,40)(27,34,35,36)(29,39,31,37);(1,4,5,3)(2,7,6,8)(10,16)(11,19)(15,17)(20,24)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,36,40,34)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=eh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=chi,d^b=de,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, 137873);