#501140

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1143 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1143 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286789
maximal normal subgroup of	#72286789

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408336);