#492877

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,g^2=hi,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321030, #72321058, #72321086, …
maximal normal subgroup of	#72321030, #72321058, #72321086, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400073);