#6508866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500434
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	#37439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2122 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(22,26)(24,32)(27,33)(28,34)(29,36)(31,35);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,22,24,26,25,28,32,34)(23,29,31,36,30,27,35,33);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23)(22,27)(24,31)(25,30)(26,33)(28,29)(32,35)(34,36);(8,16)(11,20)(13,15)(17,19);(7,14)(10,18)(13,15)(17,19)(22,28)(23,30)(26,34)(31,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314894, #72314897, #72314909, …
maximal normal subgroup of	#72314894, #72314897, #72314909, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6416062);