#428107

order	512
nilpotency class	3
isoclinism	isoclinic class of #417802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1166 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)(6,7)(9,10)(11,22)(12,27)(13,20)(14,19)(15,21)(16,24)(17,29)(18,26)(23,33)(25,32)(28,36)(30,35)(31,40)(34,38)(37,39);(1,2,4,5)(3,7,8,6)(9,11,13,24,15,26,14,25)(10,16,19,22,21,32,20,18)(12,23,28,38,30,40,29,39)(17,33,35,37,36,31,27,34);(9,12)(10,17)(11,23)(13,29)(14,28)(15,30)(16,31)(18,34)(19,27)(20,35)(21,36)(22,37)(24,39)(25,38)(26,40)(32,33);(1,3,4,8)(2,6,5,7)(9,11)(10,18)(12,23)(13,25)(14,24)(15,26)(16,20)(17,34)(19,32)(21,22)(27,33)(28,39)(29,38)(30,40)(31,35)(36,37)

Polycyclic presentation:

2^9|b^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,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, 335303);