#116385

order	512
nilpotency class	3
isoclinism	isoclinic class of #116380
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

607 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23581);