#148357

order	512
nilpotency class	4
isoclinism	isoclinic class of #135928
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 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)(6,12,15,28)(8,9)(10,24)(13,27,30,17)(14,16,33,32)(18,22)(19,35)(20,26)(23,25)(29,34,36,31);(5,6,8,12,10,16,9,14,11,15,23,28,24,32,25,33)(7,17,18,13,20,29,22,31,21,27,35,30,26,36,19,34);(2,4)(5,7)(6,13)(8,22)(9,18)(10,26)(11,21)(12,27)(14,29)(15,30)(16,34)(17,28)(19,23)(20,24)(25,35)(31,32)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224903, #72226807, #72226816, …
maximal normal subgroup of	#72224903, #72226807, #72226816, …
maximal quotient of	#72224902, #72237233, C₂×#148357

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55553);