#354455

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214887, #72311884, #72312228, …
maximal normal subgroup of	#72214887, #72311884, #72312228, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261651);