#452735

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

895 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:

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

Polycyclic presentation:

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

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, 359931);