#364198

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1456 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:

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

Polycyclic presentation:

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

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, 271394);