#421531

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

929 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)(5,6,11,15)(7,16,19,29)(8,20,23,13)(9,24,14,10)(12,25,27,18)(17,31,33,30)(21,22,28,36)(26,32,34,35);(5,7)(6,12)(8,17,23,33)(9,16)(10,25)(11,19)(13,26,20,34)(14,29)(15,27)(18,24)(21,30,28,31)(22,35,36,32);(5,8,10,22,11,23,24,36)(6,13,14,28,15,20,9,21)(7,17,18,32,19,33,25,35)(12,26,16,30,27,34,29,31);(2,4)(5,9,10,6,11,14,24,15)(7,12,18,16,19,27,25,29)(8,21,22,13,23,28,36,20)(17,26,32,30,33,34,35,31)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,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, 328727);