#397185

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60425	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1453 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)(7,19,23,36)(8,25,27,34)(9,28)(10,16)(11,17)(12,18)(13,22,32,24)(14,26,33,21)(15,30)(20,31)(29,35);(1,2)(3,4)(5,7,11,22,12,23,10,24)(6,13,17,19,18,32,16,36)(8,20,26,29,27,9,21,30)(14,31,25,35,33,28,34,15);(2,4)(5,8,10,26,12,27,11,21)(6,14,16,25,18,33,17,34)(7,20,22,30,23,9,24,29)(13,31,19,15,32,28,36,35);(5,9,12,20)(6,15,18,35)(7,21,23,26)(8,24,27,22)(10,29,11,30)(13,14,32,33)(16,28,17,31)(19,25,36,34)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,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, 304381);