#106078

order	512
nilpotency class	3
isoclinism	isoclinic class of #8534
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1116 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:

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

Polycyclic presentation:

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

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, 13274);