#238695

order	512
nilpotency class	3
isoclinism	isoclinic class of #234066
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1086 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 210 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=cg,g^b=gi,d^c=dhi,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145891);