#415115

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

856 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322311);