#211153

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
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

700 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)(11,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(6,9)(10,11)(13,15,16,23,18,25,17,24)(14,19,26,28,21,27,20,22);(6,10)(8,12)(9,11)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,28)(25,27);(1,3,4,2)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)

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,d^c=dh,f^c=fi

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, 118349);