#251398

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

875 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)(3,7)(4,6)(5,8)(10,15)(11,18)(12,14)(16,23)(19,20)(22,24)(25,26)(27,34)(28,32)(29,31)(30,35)(33,37)(36,40)(38,39);(1,3,4,8)(2,5,6,7)(9,10)(11,16)(12,21)(13,15)(14,17)(18,24)(19,23)(20,22)(25,27)(26,30)(28,35)(29,38)(31,40)(32,34)(33,39)(36,37);(9,11,13,20)(10,16,15,22)(12,18,14,19)(17,23,21,24)(26,31)(29,37)(30,39)(32,33)(34,40)(36,38);(2,6)(5,7)(9,12,13,14)(10,17,15,21)(11,19,20,18)(16,24,22,23)(26,32)(27,35)(31,33)(36,38)

Polycyclic presentation:

2^9|b^2=eh,c^2=hi,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158594);