#107392

order	512
nilpotency class	3
isoclinism	isoclinic class of #107386
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

961 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14588);