#118090

order	512
nilpotency class	3
isoclinism	isoclinic class of #118085
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

538 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25286);