#6687591

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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,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	#72282891, #72283133, #72283309, …
maximal normal subgroup of	#72282891, #72283133, #72283309, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594787);