#381469

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1392 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288665);