#286350

order	512
nilpotency class	3
isoclinism	isoclinic class of #282171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

556 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)(3,6)(4,5)(7,8)(9,10,13,16,14,17,12,22)(11,15,20,23,21,24,19,18)(25,26)(27,31)(28,33)(29,32)(30,34)(35,40)(36,38)(37,39);(2,5)(6,8)(9,11,13,20,14,21,12,19)(10,15,16,23,17,24,22,18)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,38)(34,40);(1,3,4,7)(2,6,5,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,19)(20,24)(21,23)(26,32)(27,35)(28,30)(31,38)(36,37)(39,40);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,24,18)(25,28,29,30)(26,33,32,34)(27,36,37,35)(31,38,39,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=e,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193546);