#105619

order	512
nilpotency class	3
isoclinism	isoclinic class of #105616
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

866 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,f^b=fi,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, 12815);