#111970

order	512
nilpotency class	3
isoclinism	isoclinic class of #111967
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

906 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)(3,9,11,15)(4,7)(5,8)(6,10,14,12)(13,16)(17,18,21,25,22,19,26,24)(20,23,28,32,29,27,30,31);(1,3)(2,6)(4,12)(5,11)(7,15)(8,14)(9,16)(10,13)(17,19)(18,22)(20,27)(21,24)(23,29)(25,26)(28,31)(30,32);(4,13)(6,14)(7,16)(10,12)(17,20,22,29)(18,23,19,27)(21,30,26,28)(24,32,25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19166);