#105912

order	512
nilpotency class	3
isoclinism	isoclinic class of #105909
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

982 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239729, #72239734, #72239779
maximal normal subgroup of	#72239729, #72239734, #72239779

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13108);