#146494

order	512
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=ei,f^b=fh,d^c=di,f^c=fi,f^d=fi

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, 53690);