#506303

order	512
nilpotency class	4
isoclinism	isoclinic class of #500599
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 413499);