#359393

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250732, #72252972, #72256869
maximal normal subgroup of	#72250732, #72252972, #72256869

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266589);