#211461

order	512
nilpotency class	3
isoclinism	isoclinic class of #19279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

726 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)(5,6,8,10)(7,12,9,11)(13,14,17,20)(15,21,23,28)(16,25,22,19)(18,26,27,24);(6,10)(11,12)(13,15,22,26,17,23,16,24)(14,18,25,28,20,27,19,21);(6,11)(7,9)(10,12)(15,23)(18,27)(21,28)(24,26);(1,3,4,2)(5,7,8,9)(6,12,10,11)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118657);