#240283

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3100	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gh,d^c=dh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147479);