#108130

order	512
nilpotency class	3
isoclinism	isoclinic class of #108122
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	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 15326);