#220914

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#3330	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1065 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)(5,6,10,12)(7,11,9,8)(13,14)(15,18)(16,19)(17,20)(21,24)(22,26)(23,27)(25,28);(6,8)(7,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,25)(24,28);(5,7,10,9)(6,11,12,8)(14,19)(15,22)(16,24)(18,25)(20,21)(26,27);(5,8)(6,7)(9,12)(10,11)(14,20)(18,26)(19,21)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128110);