#246153

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1331 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)(7,15)(8,11)(9,19)(10,14)(12,17)(13,16)(18,20)(26,32)(29,35);(1,2)(3,4)(5,7,10,17)(6,11,14,20)(8,9,18,16)(12,19,15,13)(21,22,25,28)(23,26,30,32)(24,27,31,34)(29,33,35,36);(2,4)(21,23,25,30)(22,26,28,32)(24,29,31,35)(27,33,34,36);(5,8,10,18)(6,12,14,15)(7,16,17,9)(11,13,20,19)(21,24)(22,27)(23,29)(25,31)(26,33)(28,34)(30,35)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153349);