#382715

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,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, 289911);