#108178

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

854 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15374);