#6372970

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60450	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2664 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)(3,4)(5,6,11,16,12,17,10,26)(7,18,9,15,21,31,20,13)(8,14,25,30,23,29,24,35)(19,32,22,28,33,36,34,27);(5,7,10,20,12,21,11,9)(6,13,16,18,17,15,26,31)(8,19,24,34,23,33,25,22)(14,27,30,32,29,28,35,36);(2,4)(5,8)(6,14)(7,19)(9,22)(10,24)(11,25)(12,23)(13,27)(15,28)(16,30)(17,29)(18,32)(20,34)(21,33)(26,35)(31,36);(5,9,11,21,12,20,10,7)(6,15,16,31,17,13,26,18)(8,22,25,33,23,34,24,19)(14,28,30,36,29,27,35,32);(8,23)(14,29)(19,33)(22,34)(24,25)(27,28)(30,35)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323138, #72323187
maximal normal subgroup of	#72323138, #72323187
maximal quotient of	#72320150

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280166);