#354459

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
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	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1512 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)(7,11)(8,18,19,12)(9,14)(10,13)(15,22,27,30)(16,23)(17,24)(20,31,32,25)(21,26)(28,33,35,36)(29,34);(5,7,9,17,10,16,21,29)(6,11,14,24,13,23,26,34)(8,15,20,28,19,27,32,35)(12,22,25,33,18,30,31,36);(5,8)(6,12)(7,15)(9,20)(10,19)(11,22)(13,18)(14,25)(16,27)(17,28)(21,32)(23,30)(24,33)(26,31)(29,35)(34,36);(1,3,4,2)(6,13)(11,23)(12,18)(14,26)(22,30)(24,34)(25,31)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280941, #72281292, #72312075, …
maximal normal subgroup of	#72280941, #72281292, #72312075, …
maximal quotient of	#72313374, #72343036

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261655);