#438348

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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)(7,15)(8,19)(9,23)(10,14)(11,17)(12,21)(13,25)(16,31)(18,30)(20,28)(22,33)(24,27)(26,32)(29,35)(34,36);(5,7,9,24,10,18,25,17)(6,11,23,15,14,27,13,30)(8,20,21,16,22,34,35,32)(12,28,33,31,29,36,19,26);(5,6)(7,11,18,27)(8,12)(9,13)(10,14)(15,24,30,17)(16,26,32,31)(19,21)(20,28,34,36)(22,29)(23,25)(33,35);(2,4)(5,8,9,21,10,22,25,35)(6,12,13,19,14,29,23,33)(7,16,17,20,18,32,24,34)(11,26,15,28,27,31,30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345544);