#246924

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
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

1058 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154120);