#353662

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251076, #72251086, #72251130, …
maximal normal subgroup of	#72251076, #72251086, #72251130, …
maximal quotient of	#72314700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260858);