#108383

order	512
nilpotency class	3
isoclinism	isoclinic class of #108372
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

737 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)(3,10)(4,13)(5,8)(6,9)(7,11)(12,16)(14,15)(17,18,22,27,23,28,21,26)(19,29,31,40,32,39,37,24)(20,33,35,25,36,41,34,42)(30,44,45,48,47,38,46,43);(1,3,5,11,6,12,4,14)(2,7,8,16,9,15,13,10)(17,19)(18,24,28,40)(20,30)(21,31)(22,37)(23,32)(25,38,42,44)(26,29,27,39)(33,43,41,48)(34,45)(35,46)(36,47);(1,2)(3,7)(4,13)(5,8)(6,9)(10,14)(11,16)(12,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,38)(26,33)(27,41)(28,42)(29,43)(31,45)(32,47)(37,46)(39,48)(40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15579);