#400720

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1405 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=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, 307916);