#107262

order	512
nilpotency class	3
isoclinism	isoclinic class of #107262
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=ei,d^c=di,e^c=eh,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, 14458);