#107632

order	512
nilpotency class	3
isoclinism	isoclinic class of #107630
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

990 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)(3,7)(4,19,22,32)(5,13)(6,18)(8,14)(9,15)(10,17)(11,21,29,25)(12,26)(16,23,28,24)(20,30,27,31);(1,3,7,15,8,18,6,17)(2,10,13,5,14,9,12,26)(4,16,21,31,23,32,27,29)(11,22,28,25,30,24,19,20);(1,4)(2,11)(3,16)(5,25)(6,27)(7,21)(8,23)(9,24)(10,22)(12,19)(13,28)(14,30)(15,31)(17,29)(18,32)(20,26)

Polycyclic presentation:

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

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, 14828);