#202787

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

832 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)(6,7)(8,9)(11,12)(13,14)(15,17)(16,24)(18,23)(19,21)(20,26)(22,25)(27,28);(1,3,4,2)(5,6,9,12,10,11,8,7)(13,15,17,14,19,23,18,21)(16,22,27,20,26,28,25,24);(5,7,8,11,10,12,9,6)(13,16,18,25,19,26,17,27)(14,20,15,22,21,24,23,28);(6,11)(7,12)(13,17,19,18)(14,15,21,23)(16,25,26,27)(20,28,24,22)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=hi,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	#72309525, #72309588, #72309594
maximal normal subgroup of	#72309525, #72309588, #72309594

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109983);