#315630

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 222826);