#107198

order	512
nilpotency class	3
isoclinism	isoclinic class of #107194
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	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,13)(12,14)(15,16)(17,18,20,24,21,23,28,31)(19,22,26,30,25,29,27,32);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19)(18,22,23,29)(20,27)(21,25)(24,32,31,30)(26,28);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(18,23)(20,28)(22,29)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14394);