#107790

order	512
nilpotency class	3
isoclinism	isoclinic class of #9303
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14986);