#412442

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1400 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)(5,6,9,14,10,15,23,13)(7,16,19,11,21,17,20,27)(8,22,24,12,25,28,36,29)(18,32,33,30,35,26,34,31);(1,2,3,4)(5,7,9,19,10,21,23,20)(6,11,13,16,15,27,14,17)(8,18,24,33,25,35,36,34)(12,26,22,30,29,32,28,31);(5,6)(7,17)(8,12)(9,13)(10,15)(11,19)(14,23)(16,21)(18,31)(20,27)(22,24)(25,29)(26,33)(28,36)(30,35)(32,34);(5,8,23,36,10,25,9,24)(6,12,14,28,15,29,13,22)(7,18,20,34,21,35,19,33)(11,26,17,31,27,32,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319638);