#106555

order	512
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

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, 13751);