#230915

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=ch,d^b=de,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, 138111);