#422926

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1931 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)(7,16)(8,17)(9,23)(10,14)(11,15)(12,19)(13,25)(18,28)(20,29)(21,32)(22,27)(24,33)(26,31)(30,34)(35,36);(1,2,3,4)(5,7,17,20,11,21,10,19)(6,12,14,32,15,29,8,16)(9,24,26,18,27,34,25,35)(13,30,22,28,31,33,23,36);(5,8,17,6,11,14,10,15)(7,12,20,32,21,29,19,16)(9,22,25,13,27,23,26,31)(18,28,24,30,35,36,34,33);(2,4)(5,9)(6,13)(7,18)(8,22)(10,26)(11,27)(12,28)(14,23)(15,31)(16,33)(17,25)(19,34)(20,24)(21,35)(29,36)(30,32)

Polycyclic presentation:

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

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, 330122);