#461149

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 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)(5,6,10,16,11,17,9,15)(7,18,26,12,22,32,21,28)(8,14,23,31,25,30,24,13)(19,27,20,29,35,36,34,33);(5,7,10,21,11,22,9,26)(6,12,16,18,17,28,15,32)(8,20,23,19,25,34,24,35)(13,29,14,27,31,33,30,36);(5,8,9,23,11,25,10,24)(6,13,15,14,17,31,16,30)(7,19,21,20,22,35,26,34)(12,27,18,29,28,36,32,33);(2,4)(5,8)(6,14)(7,20)(9,24)(10,23)(11,25)(12,27)(13,15)(16,31)(17,30)(18,33)(19,21)(22,34)(26,35)(28,36)(29,32)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,e^d=ei,g^d=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, 368345);