#202764

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ei,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^c=dh,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109960);