#364554

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2418 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271750);