#425025

order	512
nilpotency class	3
isoclinism	isoclinic class of #417690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1093 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332221);