#500113

order	512
nilpotency class	4
isoclinism	isoclinic class of #500112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1062 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307037, #72307041, #72307056
maximal normal subgroup of	#72307037, #72307041, #72307056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407309);