#220966

order	512
nilpotency class	3
isoclinism	isoclinic class of #19883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 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)(6,10)(7,8)(11,12)(13,14)(15,19)(16,23)(17,26)(18,21)(20,22)(24,28)(25,27);(5,6)(7,10)(8,12)(9,11)(13,15,18,17)(14,19,21,26)(16,24,25,22)(20,23,28,27);(13,16,18,25)(14,20,21,28)(15,22,17,24)(19,27,26,23);(5,7)(8,9)(10,12)(14,21)(19,26)(20,28)(23,27)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fi,e^d=eh

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, 128162);