#493137

order	512
nilpotency class	4
isoclinism	isoclinic class of #33404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

738 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=hi,h^2=i,b^a=be,g^a=gi,e^b=eh,f^b=fhi,h^b=hi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237237, #72239482, #72239487, …
maximal normal subgroup of	#72237237, #72239482, #72239487, …
maximal quotient of	#72319075, #72319960, C₂×#493137

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400333);