#347794

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

959 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 959 conjugacy classes of subgroups — too many to draw. Only its 174 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=di,d^c=dh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254990);