#120357

order	512
nilpotency class	3
isoclinism	isoclinic class of #120357
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

427 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,6)(4,8)(5,7)(10,16,13,14)(12,15)(17,18)(19,29)(20,34,37,42)(21,27)(22,26)(23,28)(24,31)(25,36,43,38)(30,46,48,39)(32,40)(33,44)(35,41,47,45);(3,10)(4,12)(6,14)(8,15)(9,16)(11,13)(17,19,21,31,23,33,22,32)(18,24,26,29,28,40,27,44)(20,35,36,30,37,47,38,48)(25,41,42,39,43,45,34,46);(1,3)(2,6)(4,13)(5,11)(7,9)(8,16)(10,12)(14,15)(17,20,21,36,23,37,22,38)(18,25,26,42,28,43,27,34)(19,30,31,47,33,48,32,35)(24,39,29,45,40,46,44,41)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fhi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27553);