#347810

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

839 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)(5,6,9,11)(7,13,15,20)(8,16,17,12)(10,18,19,14)(23,29)(24,31)(26,33)(28,35)(30,32)(34,36);(5,7,8,14,9,15,17,18)(6,10,12,13,11,19,16,20)(21,22,25,27)(23,26,30,34)(24,28,31,35)(29,33,32,36);(5,6)(7,10)(8,12)(9,11)(13,14)(15,19)(16,17)(18,20)(21,23,25,30)(22,26,27,34)(24,32,31,29)(28,36,35,33);(2,4)(6,11)(10,19)(12,16)(13,20)(22,27)(26,34)(28,35)(33,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=di,d^c=dh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255006);