#216106

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#2619	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

829 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cei,d^b=dh,d^c=di,f^c=fh,g^c=gh

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, 123302);