#382714

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289910);