#380994

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1157 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288190);