#470949

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

687 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fh,c^2=f,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, 378145);