#6684610

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
prime radical	C₂×#3306	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1921 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:

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=f,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	#72307791, #72308000, #72308249
maximal normal subgroup of	#72307791, #72308000, #72308249

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591806);