#458130

order	512
nilpotency class	3
isoclinism	isoclinic class of #458129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

851 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)(5,6,12,16)(7,17,20,32)(8,21,26,31)(9,10,15,11)(13,27,29,19)(14,24,23,25)(18,34,36,28)(22,30,35,33);(5,7,10,19,12,20,11,27)(6,13,9,17,16,29,15,32)(8,22,25,18,26,35,24,36)(14,30,21,28,23,33,31,34);(5,8,10,25,12,26,11,24)(6,14,9,21,16,23,15,31)(7,18,19,35,20,36,27,22)(13,28,17,33,29,34,32,30);(2,4)(5,9,10,16,12,15,11,6)(7,17,19,29,20,32,27,13)(8,23,24,21,26,14,25,31)(18,34,22,33,36,28,35,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365326);