#361451

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268647);