#106525

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

930 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)(17,18,22,26,23,27,21,35)(19,28,32,38,31,43,30,24)(20,25,33,40,34,41,36,39)(29,44,45,37,46,42,47,48);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19)(18,24,27,38)(20,29)(21,30)(22,32)(23,31)(25,37,41,48)(26,28,35,43)(33,45)(34,46)(36,47)(39,44,40,42);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,37)(26,40)(27,41)(28,42)(30,47)(31,46)(32,45)(35,39)(38,48)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=eh,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, 13721);