#115946

order	512
nilpotency class	3
isoclinism	isoclinic class of #99923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23142);