#107547

order	512
nilpotency class	3
isoclinism	isoclinic class of #107542
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

996 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:

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

Polycyclic presentation:

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

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, 14743);