#106657

order	512
nilpotency class	3
isoclinism	isoclinic class of #106654
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

900 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)(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);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(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);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(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,c^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,f^b=fh,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, 13853);