#6370093

order	512
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#575	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2962 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226779, #72239527, #72239799, …
maximal normal subgroup of	#72226779, #72239527, #72239799, …
maximal quotient of	#72343957

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277289);