#473933

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,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, 381129);