#6687439

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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 ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1943 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22)(23,30)(24,34)(25,28)(26,29)(27,33)(31,35)(32,36);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,32)(25,33)(26,31)(28,30)(29,36)(34,35);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(22,28)(23,31)(24,26)(27,35)(30,36)(32,33);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,24,25,26)(22,29,28,34)(23,32,33,31)(27,36,30,35);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(23,32)(24,26)(27,35)(29,34)(30,36)(31,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307952, #72308000, #72308405
maximal normal subgroup of	#72307952, #72308000, #72308405

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594635);