#193439

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1380 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)(3,4)(5,6,9,11,8,7,10,12)(14,18,20,23)(19,25,26,28);(5,7,9,12,8,6,10,11)(13,14,16,23,17,20,24,18)(15,19,22,28,21,26,27,25);(2,4)(6,7)(11,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(15,21)(19,26)(22,27)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,g^2=i,b^a=be,c^a=cf,c^b=ci,e^b=eh,g^b=gh,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, 100635);