#462137

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

713 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)(5,6,12,16,11,17,10,15)(7,8,19,23,21,24,20,13)(9,14,28,32,27,33,26,31)(18,22,30,25,35,36,34,29);(5,7,10,19,11,21,12,20)(6,13,16,24,17,23,15,8)(9,18,26,30,27,35,28,34)(14,29,32,36,33,25,31,22);(5,8,12,23,11,24,10,13)(6,7,15,20,17,21,16,19)(9,25,28,36,27,29,26,22)(14,30,31,18,33,34,32,35);(2,4)(5,9,10,26,11,27,12,28)(6,14,15,31,17,33,16,32)(7,18,19,30,21,35,20,34)(8,22,23,25,24,36,13,29)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=fi,d^2=e,e^2=i,f^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, 369333);