#419044

order	512
nilpotency class	3
isoclinism	isoclinic class of #419041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2298 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=i,e^2=i,f^2=i,g^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,g^b=gi,f^c=fi,g^c=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287270, #72287286, #72287360, …
maximal normal subgroup of	#72287270, #72287286, #72287360, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326240);