#506389

order	512
nilpotency class	4
isoclinism	isoclinic class of #500848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321154, #72321372, #72321417
maximal normal subgroup of	#72321154, #72321372, #72321417

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413585);