#106048

order	512
nilpotency class	3
isoclinism	isoclinic class of #106040
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

708 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,d^b=di,e^b=ei,d^c=di,e^c=ei,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, 13244);