#465977

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

861 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)(9,10,15,20)(11,21,12,17)(13,27,31,18)(14,32,24,19)(16,26,34,23)(22,39,25,35)(28,33,40,36)(29,37,30,38);(1,3,5,4)(2,6,7,8)(9,11,14,23,15,12,24,26)(10,16,19,21,20,34,32,17)(13,22,30,28,31,25,29,40)(18,33,38,39,27,36,37,35);(2,7)(6,8)(9,12)(10,17)(11,15)(13,28)(14,23)(16,32)(18,36)(19,34)(20,21)(22,30)(24,26)(25,29)(27,33)(31,40)(35,37)(38,39);(1,4,5,3)(2,8,7,6)(9,13)(10,18)(11,22)(12,25)(14,30)(15,31)(16,33)(17,35)(19,38)(20,27)(21,39)(23,28)(24,29)(26,40)(32,37)(34,36)

Polycyclic presentation:

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

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, 373173);