#6369826

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=i,h^2=i,b^a=bf,h^a=hi,f^b=fi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311278, #72311324, #72311478, …
maximal normal subgroup of	#72311278, #72311324, #72311478, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277022);