#461205

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 368401);