#470930

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

703 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:

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

Polycyclic presentation:

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

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, 378126);