#109250

order	512
nilpotency class	3
isoclinism	isoclinic class of #9784
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

646 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=h,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16446);