#251231

order	512
nilpotency class	3
isoclinism	isoclinic class of #249079
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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1132 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)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19)(25,26)(27,30)(28,31)(29,32)(33,38)(34,37)(35,39)(36,40);(1,3,4,8)(2,5,6,7)(25,27)(26,30)(28,33)(29,34)(31,38)(32,37)(35,40)(36,39);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,28)(26,31)(27,33)(29,35)(30,36)(32,39)(34,40)(37,38);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(28,35)(31,39)(33,40)(36,38)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158427);