#391526

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

737 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)(5,6,11,16,12,15,10,7)(8,17,20,13,21,28,24,27)(9,14,22,31,23,30,25,18)(19,32,35,26,36,33,34,29);(5,7,11,15,12,16,10,6)(8,13,20,17,21,27,24,28)(9,14,22,18,23,30,25,31)(19,32,35,29,36,33,34,26);(5,8,11,20,12,21,10,24)(6,13,16,28,15,27,7,17)(9,19,22,35,23,36,25,34)(14,29,31,32,30,26,18,33);(5,9,12,23)(6,14,15,30)(7,18,16,31)(8,19,21,36)(10,25,11,22)(13,26,27,29)(17,32,28,33)(20,35,24,34)

Polycyclic presentation:

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

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, 298722);