#493186

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

670 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=ci,d^a=di,g^a=gi,d^b=dh,e^b=ei,d^c=di,g^d=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400382);