#202785

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=dh,f^c=fhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309530, #72309580, #72309594
maximal normal subgroup of	#72309530, #72309580, #72309594

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109981);