#108287

order	512
nilpotency class	3
isoclinism	isoclinic class of #108282
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

720 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,e^b=ei,d^c=di,e^c=ei,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15483);