#416437

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

914 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=h,e^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=di,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, 323633);