#110552

order	512
nilpotency class	3
isoclinism	isoclinic class of #110532
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

624 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^c=eh,f^c=fh

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, 17748);