#105993

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

958 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13189);