#347804

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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 815 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)(22,26)(23,30)(24,27)(29,34)(31,32)(33,35);(5,7,14,18,9,16,8,15)(6,10,17,20,11,19,12,13)(21,22,27,34,25,29,24,26)(23,28,30,35,31,36,32,33);(5,6)(7,10)(8,12)(9,11)(13,15)(14,17)(16,19)(18,20)(21,23,25,31)(22,28,29,36)(24,32,27,30)(26,33,34,35);(2,4)(6,11)(10,19)(12,17)(13,20)(22,29)(23,31)(26,34)(30,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=i,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, 255000);