#246474

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1201 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,e^c=ei,f^c=fi,e^d=eh,f^d=fi

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, 153670);