#6677350

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256954, #72256970, #72257027, …
maximal normal subgroup of	#72256954, #72256970, #72257027, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584546);