#493217

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216211, #72216232, #72216234, …
maximal normal subgroup of	#72216211, #72216232, #72216234, …
maximal quotient of	D₈×#346

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400413);