#116213

order	512
nilpotency class	3
isoclinism	isoclinic class of #116213
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

719 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 23409);