#106353

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

996 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)(14,18)(16,23)(19,27)(20,24)(21,25)(26,28);(5,6,7,10,8,9,11,12)(13,14)(15,19)(16,24)(17,20)(18,23)(21,26)(22,28)(25,27);(6,9)(10,12)(13,15,17,22)(14,19,20,28)(16,25,23,21)(18,26,24,27)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=ci,d^b=dh,e^b=eh,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13549);