#106125

order	512
nilpotency class	3
isoclinism	isoclinic class of #8610
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

838 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18)(19,23)(20,30,31,24)(21,26)(22,25)(27,39,40,34)(28,36)(29,35)(32,43,44,37)(33,38)(41,47,48,45)(42,46);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19,21,28,22,29,33,42)(18,23,26,36,25,35,38,46)(20,27,32,41,31,40,44,48)(24,34,37,45,30,39,43,47);(3,10)(6,14)(9,15)(11,16)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,30)(26,37)(28,41)(29,40)(33,44)(35,39)(36,45)(38,43)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=ch,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13321);