#106072

order	512
nilpotency class	3
isoclinism	isoclinic class of #8516
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

733 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:

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

Polycyclic presentation:

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

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, 13268);