#246788

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1106 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153984);