#419031

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60366	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2233 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)(7,16,18,34)(8,19,21,12)(9,13)(10,25,27,14)(17,33)(20,28)(22,36)(23,32)(24,31)(26,30)(29,35);(1,2)(3,4)(5,7,10,12,11,18,27,19)(6,8,14,16,15,21,25,34)(9,22,23,35,24,28,26,17)(13,29,30,36,31,33,32,20);(2,4)(5,8,11,21)(6,12,15,19)(7,14,18,25)(9,20,24,36)(10,16,27,34)(13,28,31,22)(17,32,35,30)(23,29,26,33);(5,9)(6,13)(7,17)(8,20)(10,26)(11,24)(12,28)(14,32)(15,31)(16,33)(18,35)(19,22)(21,36)(23,27)(25,30)(29,34)

Polycyclic presentation:

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

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, 326227);