#264024

order	512
nilpotency class	3
isoclinism	isoclinic class of #264023
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 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)(3,6)(4,8)(5,7)(10,15)(12,25)(13,27)(16,30)(17,33)(18,19)(20,32)(21,37)(22,23)(26,38)(28,39)(36,40);(1,2,4,8)(3,7,5,6)(9,10,13,15,14,18,27,19)(11,20,22,32,24,30,23,16)(12,17,25,29,28,33,39,35)(21,36,37,34,26,40,38,31);(2,6)(3,5)(7,8)(9,11)(10,16)(12,26)(13,22)(14,24)(15,20)(17,34)(18,32)(19,30)(21,28)(23,27)(25,38)(29,40)(31,33)(35,36)(37,39);(1,3,4,5)(2,7,8,6)(9,12,27,39,14,28,13,25)(10,17,19,35,18,33,15,29)(11,21,23,38,24,26,22,37)(16,31,30,40,32,34,20,36)

Polycyclic presentation:

2^9|b^2=e,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=ghi,d^c=dhi,f^c=fi,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, 171220);