#361248

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

716 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)(7,17)(8,22,25,28)(9,14)(10,15)(11,16)(12,20)(13,24,29,23)(18,32,33,26)(19,30)(21,27)(31,35,36,34);(1,2)(3,4)(5,7,9,20,11,19,10,21)(6,12,15,17,16,27,14,30)(8,18,23,34,25,33,24,35)(13,26,22,31,29,32,28,36);(2,4)(5,8,9,23,11,25,10,24)(6,13,15,22,16,29,14,28)(7,18,20,34,19,33,21,35)(12,26,17,31,27,32,30,36);(6,14,16,15)(7,19)(12,17,27,30)(13,28,29,22)(18,33)(20,21)(26,31,32,36)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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	#72240851, #72252980, #72256865
maximal normal subgroup of	#72240851, #72252980, #72256865

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268444);