#105828

order	512
nilpotency class	3
isoclinism	isoclinic class of #8516
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1122 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13024);