#359596

order	512
nilpotency class	3
isoclinism	isoclinic class of #214656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3487	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3834 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)(3,4)(5,6,10,16,11,15,8,13)(7,17,18,26,20,12,24,28)(9,22,21,30,23,14,25,29)(19,27,34,35,33,31,32,36);(2,4)(5,7)(6,12)(8,18)(9,19)(10,24)(11,20)(13,28)(14,27)(15,17)(16,26)(21,32)(22,31)(23,33)(25,34)(29,36)(30,35);(5,8,11,10)(6,13,15,16)(7,18,20,24)(9,21,23,25)(12,26,17,28)(14,29,22,30)(19,32,33,34)(27,35,31,36);(2,4)(5,9,11,23)(6,14,15,22)(7,19,20,33)(8,21,10,25)(12,27,17,31)(13,29,16,30)(18,32,24,34)(26,35,28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303273, #72303359, #72318131, …
maximal normal subgroup of	#72303273, #72303359, #72318131, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266792);