#6685764

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	#39963	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,34)(32,35)(33,36);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,24,31)(22,26,27,29)(25,30,33,32)(28,35,36,34);(1,2,3,4)(22,27)(23,30)(25,33)(26,34)(29,35)(31,32);(6,11)(10,13)(12,19)(18,20)(23,31)(26,29)(30,32)(34,35);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,27)(23,32)(25,33)(26,35)(29,34)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592960);