#148125

order	512
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#486
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2041
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,d^c=di,e^c=ehi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249886, #72249894, #72249938, …
maximal normal subgroup of	#72249886, #72249894, #72249938, …
maximal quotient of	#72265623, #72280343

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55321);