#246059

order	512
nilpotency class	3
isoclinism	isoclinic class of #245999
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1350 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153255);