#148844

order	512
nilpotency class	4
isoclinism	isoclinic class of #137655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 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)(5,6,10,16,9,15,25,14,11,17,26,13,24,34,8,23)(7,12,21,30,18,31,35,28,22,32,36,33,20,29,19,27);(1,2,4,3)(5,7)(6,12)(8,19)(9,18)(10,21)(11,22)(13,33)(14,28)(15,31)(16,30)(17,32)(20,24)(23,27)(25,35)(26,36)(29,34);(1,3,4,2)(6,13)(7,18)(8,10)(9,24)(12,27)(14,15)(16,17)(19,35)(20,22)(23,34)(25,26)(28,32)(29,30)(31,33)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=ghi,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, 56040);