#193990

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1440 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101186);