#360969

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239861, #72239863, #72240851, …
maximal normal subgroup of	#72239861, #72239863, #72240851, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268165);