#419006

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2355 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)(5,6,9,16,11,17,10,15)(7,18,21,12,22,30,27,31)(8,14,25,33,26,32,24,13)(19,34,23,36,20,28,35,29);(5,7,9,21,11,22,10,27)(6,12,15,18,17,31,16,30)(8,23,24,19,26,35,25,20)(13,29,14,34,33,36,32,28);(5,8)(6,13)(7,19)(9,24)(10,25)(11,26)(12,28)(14,15)(16,32)(17,33)(18,29)(20,22)(21,35)(23,27)(30,36)(31,34);(2,4)(5,8)(6,14)(7,20)(9,25)(10,24)(11,26)(12,29)(13,15)(16,33)(17,32)(18,28)(19,22)(21,35)(23,27)(30,34)(31,36)

Polycyclic presentation:

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

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, 326202);