#253885

order	512
nilpotency class	3
isoclinism	isoclinic class of #253843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#25773	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161081);