#239048

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂×#3169	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,14)(8,10)(9,13)(11,15)(12,17)(16,19)(18,20)(21,22)(23,31)(24,28)(25,29)(26,30)(27,34)(32,35)(33,36);(6,10)(7,15)(8,17)(11,19)(12,13)(14,18)(21,22)(23,32)(24,28)(25,29)(26,30)(27,33)(31,35)(34,36);(2,4)(5,7,8,16,9,15,17,20)(6,11,12,19,13,14,10,18)(21,23)(22,27)(24,33)(25,34)(26,35)(28,31)(29,32)(30,36);(21,24,26,25)(22,28,30,29)(23,33,35,34)(27,31,36,32)

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,f^c=fhi,g^c=ghi

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, 146244);