#412290

order	512
nilpotency class	3
isoclinism	isoclinic class of #411625
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,e^c=ei

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, 319486);