#118817

order	512
nilpotency class	3
isoclinism	isoclinic class of #10508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

529 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:

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

Polycyclic presentation:

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

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, 26013);