#511963

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

954 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323044
maximal normal subgroup of	#72323044

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419159);