#197710

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

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, 104906);