#206230

order	512
nilpotency class	3
isoclinism	isoclinic class of #19651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1098 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113426);