#382456

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2466 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)(3,4)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(27,35)(28,29)(31,38)(33,34)(36,37)(39,40);(5,6)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(2,4)(6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,27,29,36,30,37,28,35)(26,31,34,40,32,39,33,38);(12,20)(15,22)(18,24)(21,23)(26,32)(31,39)(33,34)(38,40)

Polycyclic presentation:

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

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, 289652);