#251412

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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	C₂×C₄²	C₂⁴
prime radical	#25602	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

901 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:

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

Polycyclic presentation:

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

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, 158608);