#361452

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1464 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=ef,e^2=i,h^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, 268648);