#107757

order	512
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

731 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)(3,12,14,4)(5,10)(6,11)(7,9)(8,15,13,16)(17,18)(19,27,30,40)(20,31,33,24)(21,25)(22,26)(23,29,38,34)(28,43)(32,41,35,39)(36,42)(37,44)(45,48)(46,47);(1,3,6,13,7,14,5,8)(2,4,11,15,9,12,10,16)(17,19)(18,23)(20,28)(21,34)(22,30)(24,37)(25,40)(26,38)(27,42)(29,36)(31,47)(32,46)(33,45)(35,44)(39,43)(41,48);(1,4)(2,8)(3,11)(5,16)(6,15)(7,12)(9,13)(10,14)(17,20)(18,24)(19,28)(21,35)(22,33)(23,37)(25,41)(26,31)(27,43)(29,46)(30,45)(32,36)(34,44)(38,47)(39,42)(40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14953);