#295422

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=i,c^2=f,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202618);