#148656

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 55852);