#473934

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

880 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:

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

Polycyclic presentation:

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

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, 381130);